THE UNIVERSITY OF CENTRAL FLORIDA PRESENTS: THE 31 ST ...€¦ · Online registration deadline: May 5, 2019 PLENARY SPEAKERS: MARIA CHUDNOVSKY Princeton University DANIEL KRç ! Masaryk

THE 31ST CUMBERLAND CONFERENCEon Combinatorics, Graph Theory and Computing

The Cumberland Conference on Combinatorics, Graph Theory and Computing is an annual conference that brings together internationally

known researchers, industrial mathematicians, computer scientists, southeastern university and college professors, post-docs, graduate and undergraduate students to discuss the latest advances in discrete mathematics and computer science. Particular efforts are made to include faculty and students from smaller institutions throughout the region. This conference is held at a different university in the Cumberland region each year in the month of May. This year, the University of Central Florida is pleased to host the Thirty-First Cumberland Conference on its Orlando campus from May 18 – 19, 2019.

MAY 18-19, 2019University of Central Florida, Orlando, FL

THE UNIVERSITY OF CENTRAL FLORIDA PRESENTS:

FOR MORE INFORMATION VISIT:sciences.ucf.edu/math/cumberland

IMPORTANT DATES: Support request deadline: April 7, 2019Abstract submission deadline: April 21, 2019Online registration deadline: May 5, 2019

PLENARY SPEAKERS:

MARIA CHUDNOVSKYPrinceton University

DANIEL KRÁĽMasaryk University & University of Warwick

PAUL SEYMOURPrinceton University

XINGXING YUGeorgia Institute of Technology

This conference is made possible by a generous grant from the National Science Foundation. Additional support is provided by the University of Central Florida.

Scientific Committee:Mark EllinghamRobin Thomas

Organizing Committee:Zi-Xia Song (Chair)Christian BosseVaidyanathan SivaramanYue Zhao

TIME Saturday, May 18, 2019

8:00amRegistration

BA1 Lobby

8:50amOpening Remarks

Room 119

9:00amPaul Seymour

Concatenating bipartite graphs Room 119

Session 1: BA1 116 Session 2: Room 121 Session 3: Room 122

10amMark Ellingham

The structure of 4-connected K2,t-minor-free graphsSongling Shan

Eulerian hypergraphs

Farid Bouya Seymour’s second neighborhood conjecture from a different

perspective

10:25amGuoli Ding

Strengthened chain theorems for different versions of 4-connectivity

Theodore Molla Transitive tournament tilings in oriented graphs with large

total degree

Lina Li The typical structure of Gallai colorings and their extremal

graphs

10:50amCoffee Break

BA1 Lobby

11:10amBogdan Oporowski

Unavoidable immersions of large 3- and 4-edge-connected graphs

Brendan Nagle Bipartite Hansel results for hypergraphs

John Engbers Extremal independent sets and colorings in k-chromatic

graphs

11:35amSarah Allred

Unavoidable induced subgraphs of large 2-connected graphsXiaofan Yuan

Nearly perfect matchings in uniform hypergraphsJingmei Zhang

On the size of (Kt,Tk)-co-critical graphs

12:00pmYoungho Yoo

The extremal functions for triangle-free graphs with excluded minors

Blake Dunshee Closed walks in graph-encoded maps

Xujun Liu Long monochromatic paths and cycles in 2-edge-colored

multipartite graphs

12:25pmLunch (provided) and Group Photo

BA1 Lobby

* All talks are held in Business Administration I (BA1)

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University of Central Florida, Orlando, Florida, U.S.A., May 18-19, 2019

31st Cumberland Conference on Combinatorics, Graph Theory & Computing

The 31st Cumberland Conference is made possible by a generous grant from the National Science Foundation. Additional support is provided by the Department of Mathematics, the College of Sciences, and the Office of Research at the University of Central Florida.

Plenary talks are 50 minutes with 5 minutes for questions and a 5 minute break at the end.

Contributed talks are 20 minutes with 2 minutes for questions and a 3 minute break at the end.

TIME Saturday, May 18, 2019

1:30pmXingxing Yu

Coloring graphs without subdivisions of K5 Room 119

Session 1: Room 116 Session 2: Room 121 Session 3: Room 122

2:30pmMartin Rolek

Double-critical graph conjecture for claw-free graphsDong Ye

Cycle traversability and diamond linkage in polyhedral mapsKatherine Perry

Rainbow spanning trees in edge-colored complete graphs

2:55pmDantong Zhu

The extremal function for K10 minorsYifan Jing

The genus of complete 3-uniform hypergraphsDonglei Yang

On short cycles in edge-colored graphs

3:20pmTanya Lueder

A complete characterization of near outer-planar graphsYingjie Qian

Polynomial method and graph bootstrap percolationRunrun Liu

DP-coloring of planar graphs

3:45pmCoffee Break

BA1 Lobby

4:05pmWarren Shull

On spanning trees with few branch vertices CancelledThomas Lewis

The independence number of a random maximal outerplanar graph

4:30pmElena Pavelescu

Hadwiger numbers of self-complementary graphsSean English

Catching robbers quickly and efficientlyRinovia Simanjuntak

On D-magic hypercubes

4:55pmAndrei Pavelescu

The complement of a linklessly embeddable graph with at least thirteen vertices is intrinsically linked

Xiaoyun Lu On king-serf pair in tournaments

Shaohui Wang The bounds of vertex Padmakar - Ivan index on k-trees

5:20pmSarah Holliday

Distinct representatives in special set families in graphs Cancelled Daniel Johnston Deranged matchings

5:45pmRupei Xu

Robertson-Seymour Theorem–the mathematical foundation of 5G network protocol

Esmeralda Nastase The complete characterization of the minimum size supertail

Negar Orangi-Fard Maximum concurrent flow problems and p-modulus

6:10pm


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Dinner (on your own)

31st Cumberland Conference on Combinatorics, Graph Theory & ComputingUniversity of Central Florida, Orlando, Florida, U.S.A., May 18-19, 2019



TIME Sunday, May 19, 2019

8:00amRegistration

BA1 Lobby

8:30amMaria Chudnovsky Detecting odd holes

Room 119

Session 1: Room 116 Session 2: Room 121 Session 3: Room 122

9:30amGexin Yu

Graph knitting and contraction-critical graphs Cancelled Margherita Maria Ferrari Insertions yielding equivalent double occurrence words

9:55amGuantao Chen Graph knitting

Ryan Dougherty Fractal hash families of higher index

Ning Ning The non-tightness of the reconstruction threshold of a 4 states

symmetric model with different in-community and out-community mutations

10:20amCoffee Break

BA1 Lobby

10:40amFangfang Zhang

Gallai-Ramsey numbers of cyclesMichael Schroeder

Embeddings in diagonally cyclic latin squaresKevin Byrnes

The maximum length of circuit codes with long bit runs

11:05amXiaofeng Gu

Rigidity in the Euclidean plane

K. T. Arasu A new cryptosystem and algebraic constructions for its key

space

Wenjian Liu Big data Information inference on an infinite tree for a 4-state

asymmetric model with community effects

11:30amDaniel Kráľ

Extremal problems concerning tournaments Room 119

12:30pmLunch (provided)

BA1 Lobby


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TIME Sunday, May 19, 2019

1:30pm

Session 1: BA1 121 Session 2: Room 122

2:30pmHwee Kim

DNA origami words and rewriting systems

Oscar Lopez An algorithmic approach to Tran Van Trung’s basic recursive

construction of t-designs

2:55pmZhengke Miao

List star edge coloring of sparse graphsMichael Engen

Universal permutations

3:20pmDaniel Gibney

On the hardness and inapproximability of recognizing wheeler graphs

Michael Reid Self-similar tilings with polyominoes, computations, questions

and conjectures

3:45pmMajid Naroozi

Clustering in popularity adjusted stochastic block modelRamachandra Rimal

Estimation in the popularity adjusted stochastic blockmodel

4:10pm


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Conference Ends Have a safe trip!

Guangming Jing The Goldberg-Seymour conjecture on the edge-coloring of multigraphs

Room 119

Saturday, 18 May 2019 9:00

BA1, Room 119

Concatenating Bipartite Graphs

Paul SeymourPrinceton University ([email protected])

Let G be a graph with vertex set the union of sets A, B, C. Suppose that every vertex in A is ad-jacent to more than one-third of the vertices in B, and every vertex in B is adjacent to more thanone-third of C. Is it necessarily true that some vertex in C can reach half of A by two-edge paths?

We don’t know. More generally, let 0 < x, y, z < 1, where each vertex in A has at least x|B|neighbours in B, and each vertex in B has at least y|C| neighbours in C. For which x, y, z isit necessarily true that some vertex in C can reach z|A| vertices in A by two-edge paths? Thistalk is a survey of what we know about this question.

Joint work with Maria Chudnovsky, Patrick Hompe, Alex Scott and Sophie Spirkl.


BA1, Room 116

The structure of 4-connected K2,t-minor-free graphs

Mark Ellingham, Vanderbilt University ([email protected])

Guoli Ding has provided a rough structure theorem for K2,t-minor free graphs for all t. As aspecial case of his theorem, 4-connected K2,t-minor-free graphs are obtained by attaching strips,consisting of two paths joined by edges with restricted crossings, to a finite set of base graphs.The first value of t where this applies in a nontrivial way is t = 5. We give a characterization of4-connected K2,5-minor-free graphs that shows that they can be obtained from a cyclic sequenceof four types of subgraph. Consequently, we can derive a generating function and asymptoticestimate for the number of nonisomorphic 4-connected K2,5-minor-free graphs of a given order.Our work extends to general t by providing a more precise description of the strips in Ding’sresult, suggesting a general asymptotic counting conjecture. (Coauthor: J. Zachary Gaslowitz)

BA1, Room 121

Eulerian hypergraphs

Songling Shan, Illinois State University ([email protected])

Let k � 3 and H be a k-uniform hypergraph. An Euler tour in H is an alternating sequencev0, e1, v1, e2, v2, · · · , vm�1, em, vm = v0 of vertices and edges in H such that each edge of Happears in this sequence exactly once and vi�1, vi 2 ei,vi�1 6= vi, for every i = 1, 2, ...,m. Loncand Naroski showed that for k � 3, the problem of determining if a given k-uniform hypergraphhas an Euler tour is NP-complete. For 1 ` k, the minimum `-degree of H = (V,E)is �`(H) = minS✓V,|S|=` |{e |S ✓ e, e 2 E}|. Sajna and Wagner showed that every 3-uniformhypergraph H = (V,E) with �2(H) � 2 admits an Euler tour. As a consequence, every k-uniformhypergraph H = (V,E) with �k�1(H) � 2 has an Euler tour. We investigate the existence ofEuler tour in k-uniform hypergraphs for k � 4 under `-degree conditions with 1 ` k � 2. Inparticular, for k � 4, we show that every k-uniform hypergraph H = (V,E) with �2(H) � k or

�k�2(H) � 4 and with |V | � k2

2 + k

2 admits an Euler tour. (Coauthor: Amin Bahmanian)

BA1, Room 122

Seymour’s second neighborhood conjecture from a di↵erent perspective

Farid Bouya, Louisiana State University ([email protected])

Seymour’s Second Neighborhood Conjecture states that every orientation of every simple graphhas at least one vertex v such that the number of vertices of out-distance 2 from v is at least aslarge as the number of vertices of out-distance 1 from it. We present an alternative statement ofthe conjecture in the language of linear algebra. (Coauthor: Bogdan Oporowski)


BA1, Room 116

Strengthened chain theorems for di↵erent versions of 4-connectivity

Guoli Ding, Louisiana State University ([email protected])

The chain theorem of Tutte states that every 3-connected graph can be constructed from a wheelWn by repeatedly adding edges and splitting vertices. It is not di�cult to prove the followingstrengthening of this theorem: every non-wheel 3-connected graph can be constructed from W4

by repeatedly adding edges and splitting vertices. In this paper we similarly strengthen severalchain theorems for various versions of 4-connectivity.

BA1, Room 121

Transitive tournament tilings in oriented graphs with large total degree

Theodore Molla, University of South Florida ([email protected])

An orientation of a simple graph is called an oriented graph, and an orientation of the completegraph is called a transitive tournament if it does not contain a directed cycle. In this talk, wewill investigate the minimum degree threshold for oriented graphs on n = mk vertices to containa collection of m vertex-disjoint copies of the transitive tournament on k vertices.As observed by Yuster, for k = 3, the Hajnal-Szemeredi Theorem implies that 5n/6 is the correctminimum degree threshold. For k = 4, we will show that the asymptotically correct minimumdegree threshold is 11n/12. We will also discuss a number of related conjectures and results.(Coauthors: Louis DeBiasio, Allan Lo, Andrew Treglown)

BA1, Room 122

The typical structure of Gallai colorings and their extremal graphs

Lina Li, University of Illinois at Urbana-Champaign ([email protected])

An edge coloring of a graph G is a Gallai coloring if it contains no rainbow triangle. We show

that the number of Gallai r-colorings of Kn is��

r

2

�+ o(1)

�2

⇣n2

⌘

. This result indicates thatalmost all Gallai r-colorings of Kn use at most 2 colors. Following a recent trend of working onvariants of Erdos-Rothchild problem, we also study the extremal behavior of Gallai colorings.Benevides, Hoppen and Sampaio conjectured that the complete graph Kn admits the largestnumber of Gallai 3-colorings among all n-vertex graphs, while for r � 4, Hoppen, Lefmannand Odermann believes that it is the complete bipartite graph Kbn/2c,dn/2e. We resolve theabove two conjectures for su�ciently large n. Our main approach is based on the hypergraphcontainer method, developed independently by Balogh, Morris and Samotij as well as by Saxtonand Thomason, together with some stability results. (Coauthor: Jozsef Balogh)


BA1, Room 116

Unavoidable immersions of Large 3- and 4-Edge-Connected Graphs

Bogdan Oporowski, Louisiana State University ([email protected])

We show that for every integer k � 2, every su�ciently large 3-edge-connected graph admits animmersion of one of the following 3-edge-connected graphs: the graph obtained from K1,k bytripling each of its edges, the graph obtained from Ck by doubling all but one of its edges, andthe graph obtained from a k-rung ladder by contracting the two extreme rungs and joining thetwo resulting vertices by an edge. We also discuss an analogous statement for 4-edge-connectedgraphs.(Coauthor: Matthew Barnes)

BA1, Room 121

Bipartite Hansel results for hypergraphs

Brendan Nagle, University of South Florida ([email protected])

For integers n � k � 2, let V be an n-element set, and let�V

k

�denote the set of all k-element

subsets of V . For disjoint subsets A,B ✓ V , we say the pair {A,B} covers an element K 2�V

k

�

if K ✓ A[B and K \ A 6= ; 6= K \ B. We say that a collection C of such pairs {A,B} covers�V

k

�if every element of

�V

k

�is covered by at least one member of C. When k = 2, such a family

is called a separating system of V , where this concept was introduced by Renyi and studied bymany authors.

Let h(n, k) denote the minimum value ofP

{A,B}2C(|A|+ |B|) among all covers C of�V

k

�. Hansel

determined the bounds dn log2 ne h(n, 2) ndlog2 ne, and Bollobas and Scott determined anexact formula for h(n, 2). We extend these results to give an exact formula for h(n, k), and to

guarantee that all optimal covers C of�V

k

�share a common degree-sequence. Our proofs follow

lines of Bollobas and Scott, together with weight-shifting arguments in a similar vein to some ofMotzkin and Straus. (Coauthor: G. Churchill)

BA1, Room 122

Extremal independent sets and colorings in k-chromatic graphs

John Engbers, Marquette University ([email protected])

Given a family of graphs, which graph in the family has the most number of proper colorings (ver-tex colorings where adjacent vertices receive di↵erent colors)? Tomescu answered this questionfor n-vertex k-chromatic graphs, and conjectured an answer for n-vertex k-chromatic connectedgraphs. Recently, Fox, He, and Manners have proved the conjecture when using exactly k colors.A color class in a proper coloring forms an independent set of vertices, or set of pairwisenon-adjacent vertices. Which graph in a family of graphs has the most number of independentsets? We present some results in the family of n-vertex k-chromatic graphs with several di↵erentconnectivity requirements for both proper colorings and independent sets.


BA1, Room 116

Unavoidable induced subgraphs of large 2-Connected graphs

Sarah Allred, Louisiana State University ([email protected])

It is well known that, for every positive integer r, every su�ciently large connected graph containsan induced subgraph isomorphic to one of Kr, K1,r, and Pr. We prove an analogous result for 2-connected graphs. In particular, we show that every su�ciently large 2-connected graph containsan induced subgraph isomorphic to one of Kr, a subdivision of K2,r with possibly an edge joiningthe two vertices of degree r, and a graph that has a well-described ladder structure. (Coauthors:Guoli Ding, Bogdan Oporowski)

BA1, Room 121

Nearly perfect matchings in uniform hypergraphs

Xiaofan Yuan, Georgia Institute of Technology ([email protected])

We prove that, for any integers k, l with k � 3 and k/2 < l k � 1, there exists a positive realµ such that, for all integers m,n satisfying

n

k� µn m

n

k� 1�

✓1�

l

k

◆⇠k � l

2l � k

⇡

and n su�ciently large, if H is a k-uniform hypergraph on n vertices and �l(H) >�n�l

k�l

��

�(n�l)�m

k�l

�, then H has a matching of size m + 1. This improves upon an earlier result of Han,

Person, and Schacht for the range k/2 < l k�1. In many cases, our result gives tight bound on�l(H) for near perfect matchings (e.g., when l � 2k/3, n ⌘ r (mod k), 0 r < k, and r + l � kwe can take m = dn/ke � 2). When k = 3, using an absorbing lemma of Han, Person, andSchacht, our proof also implies a result of Kuhn, Osthus, and Treglown (and, independently, ofKhan) on perfect matchings in 3-uniform hypergraphs. (Coauthors: Hongliang Lu, Xingxing Yu)

BA1, Room 122

On the size of (Kt, Tk)-co-critical graphsJingmei Zhang, University of Central Florida ([email protected])

Given an integer r � 1 and graphs G,H1, . . . , Hr, we write G ! (H1, . . . , Hr) if every r-coloringof the edges of G contains a monochromatic copy of Hi in color i for some i 2 {1, . . . , r}. A non-complete graph G is (H1, . . . , Hr)-co-critical if G 9 (H1, . . . , Hr), but G+ e ! (H1, . . . , Hr) forevery edge e in G. Motivated by Hanson and Toft’s conjecture, we study the minimum numberof edges over all (Kt, Tk)-co-critical graphs on n vertices, where Tk denotes the family of all treeson k vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab.Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily Kt-saturated graph to prove that for all t � 4 and k � max{6, t}, there exists a constant c(t, k) suchthat, for all n � (t� 1)(k � 1) + 1, if G is a (Kt, Tk)-co-critical graph on n vertices, then

e(G) �✓4t� 9

2+

1

2

⇠k

2

⇡◆n� c(t, k).

Furthermore, this linear bound is asymptotically best possible when t 2 {4, 5} and k � 6. Themethod we developed may shed some light on attacking Hanson and Toft’s conjecture.(Coauthor: Zi-Xia Song)


BA1, Room 116

The extremal functions for triangle-free graphs with excluded minors

Youngho Yoo, Georgia Institute of Technology ([email protected])

Linklessly embeddable graphs are 3-dimensional analogues of planar graphs which include apexplanar graphs. While there is no known analogue of Euler’s formula for linkless embeddings, atight bound of 4n� 10 on the number of edges in linklessly embeddable graphs can be obtainedfrom their excluded minor characterization and a theorem of Mader on the extremal functions forgraphs with no Kp minor for small p. We prove an analogue of Mader’s theorem for triangle-freegraphs, and also show that apex planar graphs satisfy the edge bound of 3n� 9 + t

3 , where t isthe number of triangles. This bound is conjectured to hold for all linklessly embeddable graphs.(Coauthor: Robin Thomas)

BA1, Room 121

Closed walks in graph-encoded maps

Blake Dunshee, Vanderbilt University ([email protected])

Deng and Jin characterized all Eulerian partial duals of a ribbon graph in terms of crossing-totaldirections of its medial graph. We use graph encoded maps (gems) to extend the results of Dengand Jin. Our main results are based on consequences of the parity of certain colors in closedwalks in gems and jewels. The parity of colors in closed walks gives us characterizations of Petrieorientable cellularly embedded graphs in terms of all-crossing directions of the medial graph. Wealso describe other correspondences between properties of an embedded graph and all-crossingdirections of its medial graph and characterize embedded graphs with bipartite medial graphs.

BA1, Room 122

Long monochromatic paths and cycles in 2-edge-colored multipartitegraphs

Xujun Liu, University of Illinois at Urbana-Champaign ([email protected])

We solve four similar problems: For every fixed s and large n, we describe all values of n1, . . . , ns

such that for every 2-edge-coloring of the complete s-partite graph Kn1,...,ns there exists amonochromatic (i) cycle C2n with 2n vertices, (ii) cycle C�2n with at least 2n vertices, (iii) pathP2n with 2n vertices, and (iv) path P2n+1 with 2n + 1 vertices. This implies a generalizationof the conjecture by Gyarfas, Ruszinko, Sarkozy and Szemeredi that for every 2-edge-coloring ofthe complete 3-partite graph Kn,n,n there is a monochromatic path P2n+1. An important toolis our recent stability theorem on monochromatic connected matchings.(Coauthors: Jozsef Balogh, Alexandr Kostochka, Mikhail Lavrov)


BA1, Room 119

Coloring graphs without subdivisions of K5

Xingxing YuGeorgia Institute of Technology ([email protected])

The Four Color Theorem states that graphs without subdivisions of K5 or K3,3 are 4-colorable.What about graphs containing no subdivisions of K5? Hajos conjectured that those graphs arealso 4-colorable. We study the structure of those graphs. We show that if a graph contains K�

4(K4 minus an edge) as a subgraph then it contains a subdivision of K5 or admits a reducibleconfiguration.

This is joint work with Q. Xie, S. Xie, and X. Yuan.


BA1, Room 116

Double-critical graph conjecture for claw-free graphs

Martin Rolek, The College of William & Mary ([email protected])

A connected graph G is double-critical if �(G�u�v) = �(G)�2 for all adjacent pairs of verticesu, v 2 V (G). The double-critical graph conjecture of Erdos and Lovasz asserts that the onlydouble-critical graphs are the complete graphs. The conjecture has already been shown to holdfor some specific graph classes, and we provide information on the current progress toward thisconjecture. We also examine this conjecture for the class of claw-free graphs, where a graph isclaw-free if it does not contain the graph K1,3 as an induced subgraph. We have shown theconjecture holds for claw-free graphs with chromatic number at most eight. Using Chudnovskyand Seymour’s structure theorem for claw-free graphs, we are then able to show that any non-complete, double-critical, claw-free graph must have independence number three.(Coauthors: Sarah Loeb, Zi-Xia Song, Gexin Yu)

BA1, Room 121

Cycle traversability and diamond linkage in polyhedral maps

Dong Ye, Middle Tennessee State University ([email protected])

Let G be a graph. An embedding of G in a closed surface is polyhedral if every face is boundedby a cycle and any two faces either do not meet or meet in an edge or a vertex. A graphpolyhedral embedded in a closed surface is called a polyhedral map. It is known that everypolyhedral map is 3-connected and the neighbors of a vertex belong to a cycle. In this talk,we will talk about some recent results on cycle traversablility and K�

4 -linkage in polyhedral maps.

BA1, Room 122

Rainbow spanning trees in edge-colored complete graphs

Katherine Perry, University of Denver ([email protected])

A spanning tree of an edge-colored graph is rainbow provided that each of its edges receives adistinct color. In 1996, Brualdi and Hollingsworth conjectured that if K2m is properly (2m� 1)-edge-colored, then the edges of K2m can be partitioned into m rainbow spanning trees, exceptwhen m = 2. In this talk, we’ll look at the history and recent results concerning this conjectureand related questions and also consider the extremal question of maximizing and minimizingthe number of rainbow spanning trees in Kn, given a special type of (n� 1)-edge-coloring whichis surjective and rainbow cycle free, called a JL-coloring.


BA1, Room 116

The extremal function for K10 minors

Dantong Zhu, Georgia Institute of Technology ([email protected])

For positive integers t and n, the maximum number of edges that an n-vertex graph with no Kt

minor can have is known as the extremal function for Kt minors. In 1968, Mader proved that forevery integer t = 1, 2, ..., 7, a graph on n � t vertices and at least (t� 2)n�

�t�12

�+1 edges has a

Kt minor. Jørgensen showed that a graph on n � 8 vertices and at least 6n�20 edges either hasa K8 minor or is isomorphic to a graph obtained from disjoint copies of K2,2,2,2,2 by identifyingcliques of size 5. Song and Thomas further generalized the results for K9 minors. The extremalfunctions for Kt minors where t 9 have been important for proving several results related toHadwiger’s conjecture. In this talk, I will discuss our work-in-progress on the extremal functionfor K10 minors. (Coauthor: Robin Thomas)

BA1, Room 121

The genus of complete 3-uniform hypergraphs

Yifan Jing, University of Illinois at Urbana-Champaign ([email protected])

In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture bydetermining the genus of every complete graph Kn. In this talk, we investigate the minimum

genus embeddings of the complete 3-uniform hypergraphs K(3)n . Embeddings of a hypergraph

H are defined as the embeddings of its associated Levi graph LH with vertex set V (H) [E(H),in which v 2 V (H) and e 2 E(H) are adjacent if and only if v and e are incident in H. We

determine both the orientable and the non-orientable genus of K(3)n when n is even. Moreover,

it is shown that the number of non-isomorphic minimum genus embeddings of K(3)n is at least

214n

2 logn(1�o(1)). (Coauthor: Bojan Mohar)

BA1, Room 122

On short cycles in edge-colored graphs

Donglei Yang, Georgia Institute of Technology ([email protected])

A fundamental question in digraph theory is to establish conditions that ensures a digraphcontains certain structures. Two well-known examples are Caccetta-Haggkvist Conjecture ondirected cycles of bounded sizes and Bermond-Thomassen Conjecture on disjoint cycles. Bothare trying to establish the minimum out-degree conditions. As an analogue, this talk is devotedto finding properly colored cycles (or PC cycles, for short) and rainbow cycles in edge-coloredgraphs under color degree conditions.Our first result is to reduce the problem of finding PC cycles of length at most r to the problemof finding directed cycles of length at most r, where r � 4. In the case of rainbow C4, we provethat every edge-colored graph on n vertices with minimum color degree at least n

3 + 24pn

contains a rainbow C4, and this lower bound is asymptotically best possible. Moreover, for anyintegers s � 2 and t � 3, we prove asymptotically tight bounds n

2 + O(n1�1/s) on minimumcolor degree forcing a PC Ks,t and a rainbow Ks,t, respectively.


BA1, Room 116

A complete characterization of near outer-planar graphs

Tanya Lueder, Louisiana Tech UniversityLouisiana State University at Alexandria ([email protected])

A graph is outer-planar (OP) if it has a plane embedding in which all of the vertices lie onthe boundary of the outer face. A graph is near outer-planar (NOP) if it is edgeless or has anedge whose deletion results in an outer-planar graph. An edge whose removal results in a nearouter-planar graph is a vulnerable edge. This talk focuses on near outer-planar (NOP) graphs.We describe the class of all such graphs in terms of a finite list of excluded graphs, in a mannersimilar to the well-known Kuratowski Theorem for planar graphs. The class of NOP graphsis not closed by the minor relation, and the list of minimal excluded NOP graphs is not finiteby the topological minor relation. Instead, we use the domination relation to define minimalexcluded near outer-planar graphs, or XNOP graphs. To complete the list of 58 XNOP graphs,we give a description of those members of this list that dominate W3 or W4, wheels with threeand four spokes, respectively.

To do this, we introduce the concepts of skeletons, joints and limbs. We find an infinite list ofpossible skeletons of XNOP graphs, as well as a finite list of possible limbs. With the list ofskeletons, we permute the edges of a skeleton with the finite list of limbs to find the completelist of XNOP graphs. In this process, we also develop algorithms in SageMath to prove the listof full-K4 XNOP graphs and prove that the list of skeletons of XNOP graphs is finite.(Coauthors: Jinko Kanno, Bogdan Oporowski)

BA1, Room 121

Polynomial method and graph bootstrap percolation

Yingjie Qian, Georgia Institute of Technology ([email protected])

We introduce a simple method for proving lower bounds for the size of the smallest percolatingset in a certain graph bootstrap process. We apply this method to determine the sizes of thesmallest percolating sets in multi-dimensional tori and multi-dimensional grids (in particularhypercubes). The former answers a question of Morrison and Noel, and the latter provides analternative and simpler proof for one of their main results.(Coauthors: Lianna Hambardzumyan, Hamed Hatami)

BA1, Room 122

DP-coloring of planar graphs

Runrun Liu, Central China Normal University ([email protected])

DP-coloring (also known as correspondence coloring) is a generalization of list coloring, whichwas introduced by Dvorak and Postle in 2017. It enables some techniques which are importantin proper coloring but not feasible in list coloring (for example, vertex identification). In thistalk, we will introduce some recent progress on DP-coloring of planar graphs.


BA1, Room 116

On spanning trees with few branch vertices

Warren Shull, Emory University ([email protected])

Hamiltonian paths, which are a special kind of spanning tree, have long been of interest in graphtheory and are notoriously hard to compute. One notable feature of a Hamiltonian path is thatall vertices have degree at most two in the path. In a tree, we call vertices of degree at least threebranch vertices. If a connected graph has no Hamiltonian path, we can still look for spanningtrees that come “close,” in particular by having few branch vertices (since a Hamiltonian pathhas none).

A conjecture of Matsuda, Ozeki, and Yamashita posits that, for any positive integer k, a con-nected claw-free n-vertex graph G must contain either a spanning tree with at most k branchvertices or an independent set of 2k+3 vertices whose degrees add up to at most n� 3. In otherwords, G has this spanning tree whenever �2k+3(G) � n � 2, where �m(G) is defined as thesmallest sum of degrees of any k-vertex independent set in G. We prove this conjecture, whichwas known to be sharp, and I will also discuss the version of this problem for graphs that aren’tclaw free, which remains unsolved. (Coauthor: Ronald Gould)

BA1, Room 121

The independence number of a random maximal outerplanar graph

Thomas Lewis, Furman University ([email protected])

A set of vertices in a graph is independent provided that no two of its members are adjacent; anindependent set of largest possible size is called a maximum independent set. The independencenumber of a graph G is the size of a maximum independent set. In this talk, we show that theexpected independence number of a randomly generated maximal outerplanar graph of order nis asymptotic to (1 � e�2)n/2. Thus, on average, a maximum independent set of such a graphcomprises (about) 42% of its vertices. We also show that there is a corresponding law of largenumbers.


BA1, Room 116

Hadwiger numbers of self-complementary graphs

Elena Pavelescu, University of South Alabama ([email protected])

The Hadwiger number of a graph G, denoted by h(G), is the order of the largest complete minorof G. We prove that for all n ⌘ 0, 1(mod 4) and any self-complementary graph G with n vertices,h(G) � b(n + 1)/2c. We also prove that for all n ⌘ 0, 1(mod 4) and b(n + 1)/2c h b3n/5c,there exists a self-complementary graph G with n vertices whose Hadwiger number is h. Wederive topological properties of self-complementary graphs. (Coauthor: Andrei Pavelescu)

BA1, Room 121

Catching robbers quickly and e�ciently

Sean English, Ryerson University ([email protected])

The game of cops and robbers is a 2-player game played on a graph in which a team of copstry to catch a moving robber. The minimum number of cops necessary to catch a robber on thegraph G is the cop number, denoted c(G). In this talk we will discuss cop-throttling, in which weare concerned with catching the robber quickly. More precisely, the capture time with k cops,denoted capt

k(G), is the length of the longest game of cops and robbers possible, assuming the

cops play optimally. The cop-throttling number is given by

thc(G) := minc(G)k|V (G)|

{k + captk(G)}.

We will briefly give background on the game of cops and robbers and throttling, and then wewill show that the cop throttling number grows sublinearly with the number of vertices of G.(Coauthors: Anthony Bonato, Jane Breen, Boris Brimkov, Joshua Carlson, Jesse Geneson, LeslieHogben, K.E. Perry, Carolyn Reinhart)

BA1, Room 122

On D-magic hypercubes

Rinovia Simanjuntak, Institut Teknologi Bandung ([email protected])

For a set of distances D, a graph G of order n is said to be D�magic if there exists a bijectionf : V ! {1, 2, . . . , n} and a constant k such that for any vertex x,

Py2ND(x) f(y) = k, where

ND(x) = {y|d(y, x) = j, j 2 D}.

In this talk we search for all sets of distances Ds, such that the hypercube is D�magic. We shallutilise well-known properties of (bipartite) distance-regular graphs to construct the D�magiclabelings. (Coauthors: Palton Anuwiksa, Akihiro Munemasa)


BA1, Room 116

The complement of a linklessly embeddable graph with at least thirteenvertices is intrinsically linked

Andrei Pavelescu, University of South Alabama ([email protected])

A simple graph is called intrinsically linked if every embedding of it into R3 contains a non-trivial link. Campbell et al. showed that a graph on n � 6 vertices and at least 4n� 9 edges isintrinsically linked because it contains a K6; this implies that for a graph G with n � 15 vertices,either G or its complement is intrinsically linked. We improve this result by proving that givenany simple non-oriented graph G with at least thirteen vertices either G or its complement isintrinsically linked. (Coauthor: Elena Pavelescu)

BA1, Room 121

On King-Serf Pair in Tournaments

Xiaoyun Lu, US Census Bureau ([email protected])

For a directed graph G, a vertex x is a king if every other vertex can be reached from x by adirected path of length at most 2 and is a serf if x can be reached from every other vertex by adirected path of length at most 2. A tournament T with vertex set partition V = X [Y is called[X,Y ]-hamiltonian crossing if two ends of every hamiltonian path must belong to di↵erent parts.For two distinct vertices x and y of T we write: A = N+(x) \ N�(y), B = N+(x) \ N+(y),C = N�(x) \N�(y) and D = N�(x) \N+(y). Furthermore, let D0 = {z 2 D|z ) B,C ) z},D1 = {z 2 D|z ) B, [z, C] 6= ;}, D2 = {z 2 D|C ) z, [B, z] 6= ;} and D3 = {z 2 D|[B, z] 6=;, [z, C] 6= ;}, where X ) Y means every vertex of X dominates every vertex of Y . We callV1 [ V2 [ . . . [ Vk = V a strong component decomposition of T if Vi ) Vj for i < j and thetournament induced by Vi is strong for 1 i k, where k > 1. Our main results are thefollowing two theorems.

Theorem 1: A tournament T /2 T4 with vertex set partition X[Y is [X,Y ]-hamiltonian crossingif and only if T has a strong component decomposition V = V1 [ . . . [ Vk such that V1 ✓ X andVk ✓ Y , or V1 ✓ Y and Vk ✓ X.

Theorem 2: Let T be a tournament and (x, y) a king-serf pair in T � e where e = xy. Thenthere exists no hamiltonian [x, y]-path in T if and only if A = {a1, a2}, D3 = ;, |D0| � 2 and thesubtournament R induced by D is a tournament R(S1, D1;S2, D2), where Si = N+(ai)\D0 fori 2 {1, 2}.

BA1, Room 122

The bounds of vertex Padmakar - Ivan index on k-trees

Shaohui Wang, Texas A&M International University ([email protected])

The Padmakar - Ivan (PI) index is a distance-based topological index and a molecular structuredescriptor, which is the sum of the number of vertices over all edges uv of a graph such that thesevertices are not equidistant from u and v. In this paper, we explore the results of PI-indicesfrom trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices onk-trees are obtained by the recursive relationships, and the corresponding extremal graphs aregiven. In addition, we determine the PI-values on some classes of k-trees and compare them,and our results extend and enrich some known conclusions.


BA1, Room 116

Distinct representatives in special set families in graphs

Sarah Holliday, Kennesaw State University ([email protected])

In 2017, Hedetniemi asked the question “for which graphs G does the indexed family {NG(v)|V 2V (G)} of open neighborhoods have a system of distinct representatives?” In 2018, we answeredthat question, and explored necessary conditions and associated parameters. Now, we move onto other special set families in graphs and examine whether they do or do not have a system ofdistinct representatives. In this talk, we look at additional directions the problem is taking usin. (Coauthors: S. Hedetniemi, P. Johnson)

BA1, Room 121

Deranged matchings

Daniel Johnston, Grand Valley State University ([email protected])

The number of derangements of and n-element set can be realized as the number of perfectmatchings in a complete bipartite graph Kn,n with a perfect matching removed. For large n,this value is approximately n!/e. A related problem is the number of perfect matchings in thecomplete graph K2n with a perfect matching removed. For large n, this value is approximately(2n � 1)!!/

pe. In this talk we discuss a common generalization of these parameters by

investigating the number of perfect matchings in certain k-partite graphs.


BA1, Room 116

Robertson-Seymour theorem–the mathematical foundation of 5G net-work protocol

Rupei Xu, The University of Texas at Dallas ([email protected])

In the past four decades, Neil Robertson and Paul D. Seymour opened a new chapter for moderngraph theory by a series of papers (more than 20 papers), in which, forbidden minors andminor-closed families provided a powerful tool to analyze graph structural properties. Thoseresults are not only very important for the development of modern graph theory, but alsoreshaped graph computational complexity and algorithm design frameworks and techniques,moreover, they are closely related to parameterized algorithm, first order logic, monadicsecond order logic and more. This paper is the first research to show the powerful engineeringapplications of those tools to meet the 5G network slicing protocol design requirements. Inthe proposed protocol LogicX, finite model checking, logical and structural properties as wellas algorithmic meta-theorem-based frameworks were applied to meet the highly dynamic anddiverse agent requests. This protocol is very e�cient with low latency, flexible network topologyreconfiguration and significant reducing the cost due to provision gap.

BA1, Room 121

The complete characterization of the minimum size supertail

Esmeralda Nastase, Xavier University ([email protected])

Let q be a prime power and let n be a positive integer. Let V = V (n, q) denote the vectorspace of dimension n over Fq . A subspace partition P of V is a collection of subspaces of V withthe property that each nonzero vector is in exactly one of the subspaces in P. Suppose thatd1, . . . , dk are the di↵erent dimensions, in increasing order, that occur in the subspace partitionP. For any integer s, with 2 s k, the ds-supertail S of P is the collection of all subspacesX 2 P such that dimX < ds. Somewhat recently, it was shown that |S| � �q(ds, ds�1), where�q(ds, ds�1) denotes the minimum number of subspaces over all subspace partitions of V (ds, q)in which the largest subspace has dimension ds�1. Moreover, it was shown that if ds � 2ds�1

and equality holds in the previous bound on |S|, then the union of the subspaces in S formsa subspace. This characterization was also conjectured to hold if ds < 2ds�1. This conjecturewas recently proved in certain cases. Using a much simpler approach, we completely settle thisconjecture. (Coauthor: Papa Sissokho)

BA1, Room 122

Maximum concurrent flow problems and p-modulus

Negar Orangi-Fard, Kansas State University ([email protected])

Maximum flow problems involve finding a feasible flow through a single-source, single-sink flownetwork that has maximum value. Multicommodity maximum flow problems are a generalizationof this that involve finding an optimal flow between multiple sink and source pairs. The maximumconcurrent flow problem is a more complex and popular variation of the multicommodity flowproblem, where we are given a set of k positive demands {d1, . . . , dk}, and we are asked to finda multicommodity flow that satisfies the demands as well as possible in a certain sense. In thecase of a single source and sink, the dual optimization problem can be written in the form of ap-modulus problem with p = 1.

Modulus provides a general framework for quantifying the richness of a family of objects, �, ona graph G = (V,E). For each � 2 � and each edge e, we associate a value N (�, e) � 0 thatmeasures the usage of e by �. Given a number 1 p < 1 and a set of positive edge weights �,the p-modulus problem is formulated as

minimize⇢

Ep,�(⇢)

subject toX

e2E

N (�, e)⇢(e) � 1 8� 2 �.

Where the minimum is taken over edge density functions ⇢ : E ! R�0, and Ep,�(⇢) is a weightedp-norm.By generalizing this connection between max flow and modulus, we will show that the maximumconcurrent flow problem also has a p-modulus interpretation, which gives insight into its solution.Moreover, by modifying this modulus problem, we introduce a family of generalizations to themaximum concurrent flow problem. We will show that:

• The concurrent 1-modulus problem is equivalent to the maximum concurrent flow problem.

• The concurrent 2-modulus problem is equivalent to a generalized e↵ective resistance problem.

• The concurrent 1-modulus problem is equivalent to is the weighted sum of graph distancemeasured with respect to edge weights ��1.

Sunday, 19 May 2019 8:30

BA1, Room 119

Detecting Odd Holes

Maria ChudnovskyPrinceton University ([email protected])

A hole in a graph is an induced cycle of length at least four; and a hole is odd if it has an oddnumber of vertices. In 2003 a polynomial-time algorithm was found to test whether a graphor its complement contains an odd hole, thus providing a polynomial-time algorithm to testif a graph is perfect. However, the complexity of testing for odd holes (without accepting thecomplement outcome) remained unknown. This question was made even more tantalizing bya theorem of D. Bienstock that states that testing for the existence of an odd hole through agiven vertex is NP-complete. Recently, in joint work with Alex Scott, Paul Seymour and SophieSpirkl, we were able to design a polynomial time algorithm to test for odd holes. In this talk wewill survey the history of the problem and the main ideas of the new algorithm.

Sunday, 19 May 2019 9:30

BA1, Room 116

Graph knitting and contraction-critical graphs

Gexin Yu, The College of William & Mary ([email protected])

For a graph G = (V,E) and S ✓ V (G), (G,S) is knitted if for each partition S1, . . . , St of Swith Si 6= ;, we can find disjoint connected subgraphs G1, . . . , Gt in G such that Si ✓ V (Gi). Agraph is k-contraction-critical if its chromatic number is k, but any proper minor has chromaticnumber at most k � 1. In this talk, we give some su�cient conditions for a graph to be knitted,and apply them to obtain connectivity properties of contraction-critical graphs.

BA1, Room 121

Insertions yielding equivalent double occurrence words

Margherita Maria Ferrari, University of South Florida ([email protected])

A double occurrence word (DOW) is a word in which every symbol appears exactly twice; twoDOWs are equivalent if one is a symbol-to-symbol image of the other. We consider subwordswhich appear twice (repeat word) or which appear once along with their reverse (return word).Such subwords generalize square and palindromic factors of DOWs, respectively. Given a DOWw, we characterize the structure of w which allows two distinct insertions of repeat/return wordsto yield equivalent DOWs. This characterization depends on the locations of the insertions andon the length of the inserted repeat/return words. When one inserted word is a repeat wordand the other is a return word, then both inserted words must be trivial (i.e., have only onesymbol). The characterization also gives a method to generate families of such words recursively.(Coauthors: Daniel A. Cruz, Natasa Jonoska, Lukas Nabergall, Masahico Saito)

Sunday, 19 May 2019 9:55

BA1, Room 116

Graph knitting

Guantao Chen, Georgia State University ([email protected])

A graph G is `-knitted if for every S ✓ V (G) with |S| = ` and any partition S1|S2| . . . |Sk of S,there exist vertex-disjoint connected subgraphs G1, G2, . . . , Gk such that Si ✓ V (Gi) for eachi = 1, 2, . . . , k. When the partitions are restricted to |S1| = |S2| = · · · = |Sk| = 2, graph G iscalled k-linked, which is a well-studied subject in graph theory. We show that every 5`-connectedgraph is `-knitted and applied it to show that if there is a counterexample to Hadwiger’sconjecture that every t-chromatic graph has a Kt-minor, then the minimum counterexample (interms of graph minors) is t/5-connected.

BA1, Room 121

Fractal hash families of higher index

Ryan Dougherty, Arizona State University ([email protected])

A perfect heterogeneous hash family (PHHF ) with index � is an array with N rows, k columns,each row i contains (at most) vi symbols, and any N ⇥ t subarray contains at least � rows withall distinct symbols; this is denoted a PHHF�(N ; k, (v1, · · · , vN ), t). When v1 = · · · = vN = v,then the hash family is homogeneous, written PHF (N ; k, v, t). Let v = max(v1, · · · , vN ). IfN < d t+1

2 e, then every PHHF has a row with no duplicate entries; otherwise if N < t, then Nis a linear function of v; and if N � t, then N grows superlinearly in v.Blackburn gives a construction of homogeneous PHF s for when N is linear in v. Colbourn,Dougherty, and Horsley extend Blackburn’s result to heterogeneous hash families, and improvemany sets of parameters with fractal ingredients; a PHHF1(t; k, (v1, · · · , vt), t) is fractal if t 2,or the deletion of any row i yields a fractal PHHF1(t � 1; k, (v1, · · · , vi�1, vi+1, · · · , vt), t � 1).We generalize their results on fractal PHHF�s for arbitrary index � � 1.(Coauthor: Charles J. Colbourn)

BA1, Room 122

The non-tightness of the reconstruction threshold of a 4 states symmet-ric model with di↵erent in-community and out-community mutations

Ning Ning, University of Washington ([email protected])

The tree reconstruction problem is to collect and analyze data at the nth level of the tree,to identify whether there is non-vanishing information of the root, as n goes to infinity. Itsconnection to the clustering problem in the setting of the stochastic block model, whichhas wide applications in machine learning and data mining, has been well established. Forthe stochastic block model, an “information-theoretically-solvable-but-computationally-hard”region, or say “hybrid-hard phase”, appears whenever the reconstruction bound is not tight ofthe corresponding reconstruction on the tree problem. Although it has been studied in numerouscontexts, the existing literature with rigorous reconstruction thresholds established are verylimited, and the problem becomes extremely challenging when the model under investigationhas 4 states (the stochastic block model with 4 communities). In this paper, we study a 4 statessymmetric model with di↵erent in-community and out-community transition probabilities, andrigorously give the conditions for the non-tightness of the reconstruction threshold.

Sunday, 19 May 2019 10:40

BA1, Room 116

Gallai-Ramsey numbers of cycles

Fangfang Zhang, University of Central Florida ([email protected])

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edgescolored di↵erently. A Gallai k-coloring is a Gallai coloring that uses at most k colors. Given agraph H and an integer k � 1, the Gallai-Ramsey number GRk(H) of H is the least positiveinteger N such that every Gallai k-coloring of the complete graph KN contains a monochromaticcopy of H. The Gallai-Ramsey numbers of even cycles on at most 12 vertices and odd cycles onat most 15 vertices are known. Recently, we completely determine the Gallai-Ramsey numbersof all cycles with multiple colors. We prove that for all k � 1,

GRk(C2n) = (n� 1)k + n+ 1 for all n � 3,

GRk(C2n+1) = n · 2k + 1 for all n � 3.

(Coauthors: Yaojun Chen, Zi-Xia Song)

BA1, Room 121

Embeddings in diagonally cyclic Latin squares

Michael Schroeder, Marshall University ([email protected])

A Latin square L is diagonally cyclic if (i+ 1, j + 1, k+ 1) 2 L whenever (i, j, k) 2 L. Note that(and we discuss why) a diagonally cyclic Latin square must be of odd order. In 2004 Gruttmullerconjectured a lower bound for n which guarantees that any partial Latin square composed of kcyclic diagonals can be embedded in a diagonally cyclic Latin square of order n, and he gaveexamples to show his bound is sharp. In a separate paper, Gruttmuller proved his conjecture fork = 2.

In this talk we first discuss how embeddings in a diagonally cyclic Latin square of order n areequivalent to extensions of a partial transversal in the Cayley table of Zn to a transversal. Nextwe give a method for extending transversals in a Cayley table of Zn by finding appropriatelycolored cycles in a corresponding proper coloring of Kn,n. We use this method to sketch a prooffor the conjecture when k = 2 and k = 3; this significantly simplifies the argument given byGruttmuller for k = 2. (Coauthors: Jaromy Kuhl, Donald McGinn)

BA1, Room 122

The maximum length of circuit codes with long bit runs

Kevin Byrnes, DuPont, Wilmington, DE ([email protected])

A circuit code of spread k is a simple cycle C in the graph of the d-dimensional hypercube I(d)with the property that for any vertices x, y 2 C, dI(d)(x, y) � min{dC(x, y), k}. One applicationof circuit codes is as error-correcting codes, so it is of interest to find the maximum length ofa circuit code in dimension d with spread k, K(d, k). However, finding closed form expressionsfor K(d, k) for classes of (d, k) combinations is extremely rare. In this talk we present recentresults building upon the work of Singleton, Douglas, and others that gives an exact formulafor K(d, k) for an infinite class of symmetric circuit codes with long bit runs (sequences ofdistinct transitions). We also present two conjectures: that this formula for K(d, k) holds foran even larger class of circuit codes, and a proposed extension of Douglas’s Isomorphism Theorem.

Sunday, 19 May 2019 11:05

BA1, Room 116

Rigidity in the Euclidean plane

Xiaofeng Gu, University of West Georgia ([email protected])

Rigidity, arising from mechanics, is the property of a structure that does not flex. A combina-torial characterization of rigidity in the Euclidean plane has been obtained by Laman in 1970,which results the following definition of rigid graphs. A graph G is sparse if |E(H)| 2|V (H)|�3for every subgraph H of G with |V (H)| � 2; if in addition |E(G)| = 2|V (G)| � 3, then G isminimally rigid. A graph is rigid if it contains a spanning minimally rigid subgraph. A theoremof Lovasz and Yemini in 1982 shows that every 6-connected graph is rigid. Some improvedresults and related results will be presented in this talk.

BA1, Room 121

A new cryptosystem and algebraic constructions for its key space

Krishnasamy Arasu, Riverside Research, Beavercreek, OH ([email protected])

In this joint work with Michael Clark, Anika Goyal and Abhishek Puri, we introduce a newsymmetric encryption scheme using snake-and-ladder blocks. While the novelty of our toplevel algorithm is on key generation for a symmetric cryptographic cipher, we instantiate thiscryptosystem using a class of combinatorial designs called “Di↵erence Set Pairs (DSPs)”. Theencryption algorithm uses a specific matrix constructed from one of the two di↵erence setsin the DSP. The decryption at the receivers end is done using the matrix constructed fromthe other di↵erence set in the DSP. The proposed encryption scheme using DSPs is moree�cient than encryption using general orthogonal matrices, as we need to store only one rowof the matrix instead of the complete 2D matrix without even compromising the orthogonalproperty. In this data-centric era, cost and ease of use have made cloud computing the chosentool for big-data processing and analytics. Problems arise with cloud computing concerningdata privacy, security and authenticity. Our new simple symmetric key algorithm could serveas an e�cient method for providing data-storage security in cloud computing. Data users canreconstruct the requested data from cloud server using a shared secret key. We suggest a hybrid(combination) encryption method of using asymmetric and symmetric cryptographic algorithms.The symmetric key distribution among cloud provider and legitimate users can be accomplishedutilizing asymmetric techniques.

BA1, Room 122

Big data information inference on an infinite tree for a 4-state asym-metric model with community e↵ects

Wenjian Liu, City University of New York ([email protected])

The tree reconstruction problem is to collect and analyze data at the nth level of the tree,to identify whether there is non-vanishing information of the root, as n goes to infinity. Itsconnection to the clustering problem in the setting of the stochastic block model, whichhas wide applications in machine learning and data mining, has been well established. Forthe stochastic block model, an “information-theoretically-solvable-but-computationally-hard”region, or say “hybrid-hard phase”, appears whenever the reconstruction bound is not tight ofthe corresponding reconstruction on the tree problem. This problem has wide applications invarious fields such as biology, information theory and statistical physics, and its close connections

to cluster learning, data mining and deep learning have been well established in recent years.It is quite challenging with the techniques used including Markov chains, statistics, statisticalphysics, information theory, cryptography and noisy computation. Although it has beenstudied in numerous contexts, the existing literatures with rigorous reconstruction thresholdsestablished are very limited. In this project, inspired by a classical deoxyribonucleic acid(DNA) evolution model, the Felsenstein-1981 model, and also taking into consideration of theCharga↵’s parity rule by allowing the existence of a guanine-cytosine content bias, we study thenoise channel in terms of an 4-state asymmetric probability transition matrix with communitye↵ects, for four nucleobases of DNA. The corresponding information reconstruction problemin molecular phylogenetics is explored, by means of refined analyses of moment recursion,in-depth concentration estimates, and thorough investigations on an asymptotic 4-dimensionalnonlinear second order dynamical system. We rigorously show that the reconstruction boundis not tight when the sum of the base frequencies of adenine and thymine falls in the interval⇣0, 1/2�

p3/6

⌘[

⇣1/2 +

p3/6, 1

⌘, which is the first rigorous result on asymmetric noisy

channels with community e↵ects.

Sunday, 19 May 2019 11:30

BA1, Room 119

Extremal Problems Concerning Tournaments

Daniel Kral’Masaryk University and University of Warwick ([email protected])

A tournament is an orientation of a complete graph. We provide a brief overview of variousextremal problems related to tournaments, particularly those concerning cycles. We particularlyfocus on the conjecture of Linial and Morgenstern that, among all tournaments with a givendensity d of cycles of length three, the density of cycles of length four is minimized by a randomblow-up of a transitive tournament with all but one parts of equal sizes, i.e., a tournamentwith the structure similar to graphs appearing in the Erdos-Rademacher problem on trianglesin graphs with a given edge density. We prove the conjecture of Linial and Morgenstern ford � 1/36 using methods from spectral graph theory. We also demonstrate that the structure ofextremal examples is more complex than expected and give its full description for d � 1/16.

The results concerning the conjecture of Linial and Morgenstern are based on joint work withTimothy Chan, Andrzej Grzesik and Jonathan Noel.

Sunday, 19 May 2019 1:30

BA1, Room 119

The Goldberg-Seymour conjecture on the edge-coloring of multigraphs

Guangming Jing1

Georgia State University ([email protected])

Given a multigraph G = (V,E), the edge-coloring problem (ECP) is to color the edges ofG with the minimum number of colors such that no two incident edges receive the samecolor. This problem can naturally be formulated as an integer program, and its linearprogramming relaxation is called the fractional edge-coloring problem (FECP). The optimalvalue of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index)of G, denoted by �0(G) (resp.�⇤(G)). Let �(G) be the maximum degree of G and let

w(G) = maxH✓G

⇢|E(H)|

b 12 |V (H)|c

�. Clearly, max{�(G), dw(G)e} is a lower bound for �0(G).

As shown by Seymour, �⇤(G) = max{�(G), dw(G)e}. In the 1970s Goldberg and Seymourindependently conjectured that �0(G) max{�(G) + 1, dw(G)e}, which if true implies that,first, every multigraph G satisfies �0(G) � �⇤(G) 1, so FECP enjoys a fascinating integerrounding property; second, ECP can be approximated within one of the optimum, and hence isone of the “easiest” NP-hard problems; third, there are only two possible values for �0(G), so ananalogue to Vizing’s theorem on edge-colorings of simple graphs holds for multigraphs. In thistalk, we will discuss a proof of this conjecture.

This is joint work with Guantao Chen and Wenan Zang.

1Special 50-minute talk by a graduate student (to graduate in 2019) .

Sunday, 19 May 2019 2:30

BA1, Room 121

DNA origami words and rewriting systems

Hwee Kim, University of South Florida ([email protected])

Self-assembly is a process where smaller components (usually molecules) autonomously assembleto form a larger structure. A well-known self-assembly variant is the DNA origami system.In DNA origami, a single-stranded DNA plasmid, called the sca↵old, outlines a shape, whileshort DNA strands, called staples, connect di↵erent parts of the sca↵old, fixing the terminalrigid structure. We classify rectangular DNA origami structures according to their sca↵old andstaples organization by associating a graphical representation to each sca↵old folding. Inspired bythe well-studied Jones monoid, we identify basic modules that form the structures. The graphicaldescription is obtained by ‘gluing’ basic modules one on top of the other. To each module weassociate a symbol such that gluing of modules corresponds to concatenating the associatedsymbols. Every word corresponds to a graphical representation of a DNA origami structure, anda set of rewriting rules defines equivalent words that correspond to the same graphical structure.We propose two di↵erent types of basic modules that describe a set of DNA origami structuresin di↵erent ways. For each type, we first find the necessary and su�cient set of rewriting rulesthat describes equivalence of graphical structures. Then, we provide the number of all possiblestructures through the number of equivalence classes of words based on the rewriting rules. Wealso give a polynomial time algorithm that computes the shortest word for each equivalence class.(Coauthors: James Garrett, Natasa Jonoska, Masahico Saito)

BA1, Room 122

An algorithmic approach to Tran Van Trung’s basic recursive construc-tion of t-designs

Oscar Lopez, Florida Atlantic University ([email protected])

In 2017 and 2018, Tran Van Trung introduced new recursive techniques to construct t� (v, k,�)designs. In this work, we enhance Tran Van Trung’s “Basic Construction” by a robust ande�cient hybrid computational apparatus which enables us to construct hundreds of thousandsof new t� (v, k,⇤) designs from previously known ingredient designs.

Sunday, 19 May 2019 2:55

BA1, Room 121

List star edge coloring of sparse graphs

Zhengke Miao, Jiangsu Normal University ([email protected])

A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraphis a path of length at most 3. The star chromatic index �0

st(G) of a graph G is the smallest

integer k such that G has a star k-edge-coloring. The list star chromatic index ch0st(G) is

defined analogously. In this talk, I will present a new upper bound on ch0st(G) for sparse graphs.

Specifically we show that for every " > 0 there exists a constant c(") such that if mad(G) < 83 �",

then ch0st(G) 3�

2 + c(") and the coe�cient 32 of � is the best possible.

(Coauthors: Katie Horacek, Jiaao Li, Rong Luo)

BA1, Room 122

Universal permutations

Michael Engen, University of Florida ([email protected])

The problem of finding a shortest permutation which contains all permutations of length n datesto 1999. While its length is known to be asymptotically quadratic in n, surprisingly little elseis known. We prove a new constructive upper bound, as well as prove a formula for the exactlength of the shortest permutation containing all layered permutations of length n.

Sunday, 19 May 2019 3:20

BA1, Room 121

On the hardness and inapproximability of recognizing Wheeler graphs

Daniel Gibney, University of Central Florida ([email protected])

In recent years the relationship between a newly defined class of graphs and several importantstring indexing structures has been discovered. This class of graphs, known as Wheeler graphs,were shown by Gagie et al. to model de Bruijn graphs, generalized compressed su�x arrays, andseveral other BWT related structures. Moreover, the Wheeler graph axioms reveal a su�cientcondition for a data structure to be indexed e�ciently. In our work, we prove the NP-hardnessof recognizing Wheeler graphs, in addition to providing an exponential time algorithm for therecognition problem which has better time complexity than the naive approach. We also showthe APX-hardness of finding the minimum number of edges that must be removed to transform agraph into a Wheeler graph. On the other hand, we demonstrate that the dual of this optimizationproblem, finding the maximal Wheeler graph, admits a constant approximation.(Coauthor: Sharma V. Thankachan)

BA1, Room 122

Self-similar tilings with polyominoes, computations, questions and con-jectures

Michael Reid, University of Central Florida ([email protected])

A polyomino is called rectifiable if it can tile a rectangle, and is called a rep-tile if it can tile alarger shape similar to itself. It is easy to show that if a polyomino is rectifiable, then it is arep-tile. Is the converse true? This has been an open question for over fifty years.In this talk, we consider the questions of finding the smallest rep-tiling of a polyomino, and findingall, or almost all rep-tilings. Numerous examples will be given, and several open problems willbe presented.

Sunday, 19 May 2019 3:45

BA1, Room 121

Clustering in popularity adjusted stochastic block model

Majid Noroozi, University of Central Florida ([email protected])

In the present talk, we consider the Popularity Adjusted Stochastic Block Model (PABM) whichhas been recently introduced by Sengupta and Chen (2018). In the PABM, the probability ofconnection between nodes is a product of popularity parameters that depend on the communitiesto which the nodes belong as well as on the pair of nodes themselves. The authors showed thatPABM generalizes both the Stochastic Block Model (SBM) and the Degree-Corrected BlockModel (DCBM) and suggested the quasi-maximum likelihood type procedure for estimation andclustering. However, the authors considered only the case of a small finite number of communities,and the spectral clustering, that they used for implementation of the modularity optimization,does not recover communities reliably when the probability of connection of nodes in the networkis very diverse. The purpose of the present talk is to address the deficiency of spectral clusteringin the latter case. In particular, we propose to use a di↵erent type of clustering for the PABMdata. Experiments on a synthetic data set demonstrate the e↵ectiveness of our approach.

BA1, Room 122

Estimation in the popularity adjusted stochastic blockmodel

Ramchandra Rimal, University of Central Florida ([email protected])

Consider a network with its adjacency matrix Aij ⇠ Ber(Pij). We estimate the probabilitymatrix P . Our estimation technique involves the penalized optimization procedure. We estimatethe matrix of the probability of connection between nodes by minimizing the squared di↵erencesbetween the blocks of the matrix A to its best rank one approximation over the set of all possibleclustering matrix. We use the oracle inequality to find the upper bound of the estimation error.(Coauthor: Marianna Pensky)

Thank you all for attending the 31st Cumberland Conference on Combinatorics, Graph Theory & Computing. Special thanks to Linda Perez-Rodriguez and Doreen Goulding for their tireless efforts to ensure our success. Have a safe trip! Organizing Committee: Zi-Xia Song Christian Bosse Vaidyanathan Sivaraman Yue Zhao

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THE UNIVERSITY OF CENTRAL FLORIDA PRESENTS: THE 31 ST ...€¦ · Online registration deadline: May 5, 2019 PLENARY SPEAKERS: MARIA CHUDNOVSKY Princeton University DANIEL KRç ! Masaryk