Generalized Balloons and Chinese Postman Problems in ...suilo2/preprint/slide...Chinese Postman Problem Motivation and Questions In 1891, Petersen proved that every cubic graph without

Motivation and QuestionsPrevious Results using Balloons

Smallest Matching Number and Edge-ConnectivityChinese Postman Problem

Generalized Balloons and Chinese Postman

Problems in Regular Graphs

Suil O and Douglas B. West

Department of MathematicsUniversity of Illinois at Urbana-Champaign

2009 SIAM Annual Meeting

O and West Generalized Balloons and Chinese Postman Problems



Table of Contents

Motivation and Questions

Previous Results using Balloons

Smallest Matching Number and Edge-Connectivity

Chinese Postman Problem





In 1891, Petersen proved that every cubic graph without cut-edgeshas a perfect matching.





In 1891, Petersen proved that every cubic graph without cut-edgeshas a perfect matching.If there are cut-edges in a cubic graph, then what happens?





In 1891, Petersen proved that every cubic graph without cut-edgeshas a perfect matching.If there are cut-edges in a cubic graph, then what happens?More generally, we consider connected (2r + 1)-regular graphs.






1) How many cut-edges can a connected (2r + 1)-regular graphwith n vertices have?






1) How many cut-edges can a connected (2r + 1)-regular graphwith n vertices have?2) How small can the matching number α′(G ) be in a connected(2r + 1)-regular graph with n vertices?






1) How many cut-edges can a connected (2r + 1)-regular graphwith n vertices have?2) How small can the matching number α′(G ) be in a connected(2r + 1)-regular graph with n vertices?3) Can we characterize when equality holds? (i.e. when is thematching number minimized?)






1) How many cut-edges can a connected (2r + 1)-regular graphwith n vertices have?2) How small can the matching number α′(G ) be in a connected(2r + 1)-regular graph with n vertices?3) Can we characterize when equality holds? (i.e. when is thematching number minimized?)4) Are there other applications of balloons?




Definitions

◮ Graphs in which every vertex has degree 3 are cubic graphs.




Definitions


◮ The matching number α′(G ) of a graph is the maximum sizeof a matching in it.




Definitions



◮ Let c(G ) denote the number of cut-edges in a graph G .




Definitions



◮ Let c(G ) denote the number of cut-edges in a graph G .

◮ Let b(G ) denote the number of balloons in a graph G .




Tools - Balloons and Br

B1 a cubic graph with two balloons

◮ A balloon in a graph G is a maximal 2-edge-connectedsubgraph of G that is incident to exactly one cut-edge of G .







◮ Br : the unique graph with 2r + 3 vertices having 2r + 2vertices of degree 2r + 1 and one vertex of degree 2r .(Br is the complement of P3 + rK2)







◮ Br : the unique graph with 2r + 3 vertices having 2r + 2vertices of degree 2r + 1 and one vertex of degree 2r .(Br is the complement of P3 + rK2)

◮ Br is the smallest possible balloon in a (2r + 1)-regular graph.




Previous Results

Fn : the family of connected (2r + 1)-regular graphs with nvertices.




Previous Results


Theorem (O and West 2009+)

If G ∈ Fn, then c(G ) ≤r(n−2)−22r2+2r−1

− 1 cut-edges, which reduces ton−73 for cubic graphs. Equality holds infinitely often.




Previous Results



If G ∈ Fn, then c(G ) ≤r(n−2)−22r2+2r−1

− 1 cut-edges, which reduces ton−73 for cubic graphs. Equality holds infinitely often.

Theorem (Henning and Yeo 2007)

If G ∈ Fn, then α′(G ) ≥ n2 −

r2

(2r−1)n+2(2r+1)(2r2+2r−1)

.




Construction - Tr and Hr

a graph in T1 a graph in H1

◮ Tr : the family of trees such that every non-leaf vertex hasdegree 2r + 1 and all the leaves have the same color in aproper 2-coloring.




Construction - Tr and Hr

a graph in T1 a graph in H1

◮ Tr : the family of trees such that every non-leaf vertex hasdegree 2r + 1 and all the leaves have the same color in aproper 2-coloring.

◮ Hr : the family of (2r + 1)-regular graphs obtained from treesin Tr by identifying each leaf of such a tree with the vertex ofdegree 2r in a copy of Br .




Properties of graphs in Hr

Proposition (O and West 2009+)

Let G be an n-vertex graph in Hr .

◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)







◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)

◮ b(G ) = (2r−1)n+24r2+4r−2







◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)

◮ b(G ) = (2r−1)n+24r2+4r−2

◮ α′(G ) = 12n −r2

(2r−1)n+2r(4r3+6r2−1)







◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)

◮ b(G ) = (2r−1)n+24r2+4r−2

◮ α′(G ) = 12n −r2

(2r−1)n+2r(4r3+6r2−1)

◮ c(G ) = r(n−2)−22r2+2r−1

− 1







◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)

◮ b(G ) = (2r−1)n+24r2+4r−2

◮ α′(G ) = 12n −r2

(2r−1)n+2r(4r3+6r2−1)

◮ c(G ) = r(n−2)−22r2+2r−1

− 1


When n is as above, G has the smallest matching number over allconnected (2r + 1)-regular graphs if and only if G is in Hr .




Balloons and Total Domination

◮ A subset S is a dominating set in a graph G if every vertexoutside S has a neighbor in S .






◮ A subset S is a total dominating set in a graph G if everyvertex in V (G ) has a neighbor in S . The total dominationnumber of G , denoted γt(G ), is the least size of such a set.







Theorem (Henning, Kang, Shan, and Yeo 2008)

When G is regular with degree at least 3, γt(G ) ≤ α′(G ).







Theorem (Henning, Kang, Shan, and Yeo 2008)

When G is regular with degree at least 3, γt(G ) ≤ α′(G ).


If G is a connected cubic graph, then γt(G ) ≤ α′(G ) − b(G )/6,

unless b(G ) = 3 and there is only one vertex outside the balloons.




New Questions

◮ What is the smallest matching number for the family oft-edge-connected (2r + 1)-regular graphs with n vertices?(Also for 2r -regular graphs.)




New Questions

◮ What is the smallest matching number for the family oft-edge-connected (2r + 1)-regular graphs with n vertices?(Also for 2r -regular graphs.)

◮ What is the best upper bound for the length of Chinesepostman tours in t-edge-connected (2r + 1)-regular graphswith n vertices?






If G is a (2t + 1)-edge-connected (2r + 1)-regular graph with nvertices, then α′(G ) ≥ n2 − (

r−t2(r+1)2+t

)n2 , and this is sharp for t ≥ 1.







r−t2(r+1)2+t


Proof. Let S be a set with maximum deficiency. Thus,α′(G ) = 12 (n − def(S)), where def(S) = o(G − S) − |S |.







r−t2(r+1)2+t



Let ci count the odd components of G − S having i edges to S .Let c = c(2t+1) + ... + c(2r−1), and let c

′ = o(G − S) − c .







r−t2(r+1)2+t



Let ci count the odd components of G − S having i edges to S .Let c = c(2t+1) + ... + c(2r−1), and let c

′ = o(G − S) − c .

Note that for (2t + 1) ≤ i ≤ 2r − 1, each odd component of G − Shaving i edges to S has at least (2r + 3) vertices.(Otherwise, each vertex of G has a neighbor outside.)




Completion of Proof

Counting the edges joining S to odd components of G − S yields(2r + 1)|S | ≥ (2r + 1)c ′ + (2t + 1)c .




Completion of Proof


Hence |S | ≥ c ′ + ( 2t+12r+1 )c ≥ (2t+12r+1 )c .




Completion of Proof


Hence |S | ≥ c ′ + ( 2t+12r+1 )c ≥ (2t+12r+1 )c .

Therefore, n ≥ |S | + c(2r + 3) ≥ ( 2t+12r+1 )c + c(2r + 3),

which implies that c ≤ ( 2r+14r2+4r+4+2t

)n. Now, we compute




Completion of Proof


Hence |S | ≥ c ′ + ( 2t+12r+1 )c ≥ (2t+12r+1 )c .

Therefore, n ≥ |S | + c(2r + 3) ≥ ( 2t+12r+1 )c + c(2r + 3),

which implies that c ≤ ( 2r+14r2+4r+4+2t

)n. Now, we compute

def(S) = (c + c ′) − |S | ≤ c −2t + 1

2r + 1c =

2(r − t)

2r + 1c

≤2(r − t)

2r + 1

(

2r + 1

4r2 + 4r + 4 + 2t

)

n =(r − t)n

2(r + 1)2 + t.




Bounds for Other Cases


If G is a 2t-edge-connected (2r + 1)-regular graph with n vertices,then again α′(G ) ≥ n2 − (

r−t2(r+1)2+t

)n2 , and this is sharp.

(same bound as when κ′(G ) ≥ 2t + 1)




Bounds for Other Cases


If G is a 2t-edge-connected (2r + 1)-regular graph with n vertices,then again α′(G ) ≥ n2 − (

r−t2(r+1)2+t

)n2 , and this is sharp.

(same bound as when κ′(G ) ≥ 2t + 1)


If G is a (2t − 1)-edge-connected 2r -regular graph with n vertices,then α′(G ) ≥ n2 −(

r−t2r2+r+t

)n2 , and this inequality is sharp even whenG is 2t-edge-connected.




Sharpness for Odd Regular Graphs - Br ,t and Gr ,t

B2,1 a (5, 3)-biregular bigraph H G2,1

◮ Let Br ,t be a graph obtained from the graph Br by deleting amatching with t edges in Br .







◮ To make a (2t + 1)-edge-connected (2r + 1)-regular Gr ,t ,replace each vertex of T in a (2t + 1)-edge-connected(2r +1, 2t +1)-biregular (S ,T )-bigraph H with a copy of Br ,t .







◮ To make a (2t + 1)-edge-connected (2r + 1)-regular Gr ,t ,replace each vertex of T in a (2t + 1)-edge-connected(2r +1, 2t +1)-biregular (S ,T )-bigraph H with a copy of Br ,t .

◮ Note: |S | = q(2t + 1) and |T | = q(2r + 1) for some q ∈ Q.




Matching Number of Gr ,t


For 0 ≤ t ≤ r , α′(Gr ,t) =n2 − (

r−t2(r+1)2+t

)n2 ,

where n = q(2t + 1) + q(2r + 1)(2r + 3) = q(

(2r + 2)2 + 2t))

.






For 0 ≤ t ≤ r , α′(Gr ,t) =n2 − (

r−t2(r+1)2+t

)n2 ,

where n = q(2t + 1) + q(2r + 1)(2r + 3) = q(

(2r + 2)2 + 2t))

.

Proof. By the previous theorem, α′(Gr ,t) ≥n2 − (

r−t2(r+1)2+t

)n2 .






For 0 ≤ t ≤ r , α′(Gr ,t) =n2 − (

r−t2(r+1)2+t

)n2 ,

where n = q(2t + 1) + q(2r + 1)(2r + 3) = q(

(2r + 2)2 + 2t))

.

Proof. By the previous theorem, α′(Gr ,t) ≥n2 − (

r−t2(r+1)2+t

)n2 .

Recall that |S | = (2t + 1)q. Thus,

def(S) = o(G − S) − |S | = q(2r + 1) − q(2t + 1)

= 2q(r − t) = 2n

(2r + 2)2 + 2t(r − t) =

(r − t)n

2(r + 1)2 + t.




Edge-Connectivity of Br ,t

Lemma

For 0 ≤ t ≤ r , the edge-connectivity of the graph Br ,t is 2r .





Lemma


Proof.(Elementary exercise) If a graph G is connected, andn2 ≤ δ(G ) ≤ n, then κ

′(G ) = δ(G ), and the above is a special case.





Lemma


Proof.(Elementary exercise) If a graph G is connected, andn2 ≤ δ(G ) ≤ n, then κ

′(G ) = δ(G ), and the above is a special case.

Lemma

Iteratively replacing a vertex of T with Br ,t in H preserves (2t +1)-edge-connectedness.




Construction of Br ,t,k

To show that equality holds infinitely often, we need to build aninfinite family of such H.

Cyclic construction using two copies of K3,5, and B2,1,2

We are gonna put a bunch of copies of K2t+1,2r+1 around a circleand modify them to construct H.




Edge Connectivity of Br ,t,k

Lemma

For a ≥ b, if a graph H is the graph obtained from Ka,b by deletinga matching of size b, then κ′(H) = b − 1.





Lemma


Proof.(Elementary exercise) If a graph G is a bipartite graph withdiameter at most 3, then κ′(G ) = δ(G ), and the above is a specialcase.





Lemma


Proof.(Elementary exercise) If a graph G is a bipartite graph withdiameter at most 3, then κ′(G ) = δ(G ), and the above is a specialcase.


For 0 ≤ t ≤ r , κ′(Br ,t,k) = 2t + 1




Sharpness for Even Regular Graphs : B ′r ,t and G′r ,t

Let B ′r ,t be the graph obtained from K2r+1 by deleting a matchingof with t edges.To make a 2t-edge-connected 2r -regular graph G ′r ,t , replace eachvertex of T in 2t-edge-connected (2r , 2t)-biregular (S ,T )-bipartiteH ′ with a copy of B ′r ,t.Similarly, we can have an infinite family of 2t-edge-connected2r -regular graphs like in previous steps.





A Chinese Postman tour in a connected graph G is a shortestclosed walk traversing all edges in G .






Let eP(G ) be the number of edges in it.







A parity subgraph in a graph G is a spanning subgraph H of Gsuch that dG (v) ≡ dH(v) (mod 2) for every vertex v in G .








Let p(G ), the parity number of G , be the minimum number ofedges in a parity subgraph of G .








Let p(G ), the parity number of G , be the minimum number ofedges in a parity subgraph of G .

Note that eP(G ) = |E (G )| + p(G ). In view of many applicationsof the Chinese Postman problem, it is natural to ask for the valueof eP(G ), or equivalently, the value of p(G ).




Construction of H′r

a graph in T ′1 a graph in H′

1

◮ Let T ′r be the family of trees such that every non-leaf vertexhas degree 2r + 1.






1


◮ Let H′r be the family of (2r + 1)-regular graphs obtained fromtrees in T ′r by identifying each leaf of such a tree with theneck in a copy of Br .






1


◮ Let H′r be the family of (2r + 1)-regular graphs obtained fromtrees in T ′r by identifying each leaf of such a tree with theneck in a copy of Br .

◮ For r = 1, we will show the graph in H′r have the largest valueof p(G ) among n-vertex cubic graphs.




Parity Number of H′r

Proposition

Let pr = 2r2 + 2r − 1. For any n-vertex graph G in H′r ,

b(G ) = (2r−1)n+22pr , c(G ) =r(n−2)−2

pr− 1.





Proposition


b(G ) = (2r−1)n+22pr , c(G ) =r(n−2)−2

pr− 1.

Lemma

If G is regular of odd degree, then every cut-edge is in every paritysubgraph.





Proposition


b(G ) = (2r−1)n+22pr , c(G ) =r(n−2)−2

pr− 1.

Lemma

If G is regular of odd degree, then every cut-edge is in every paritysubgraph.

Corollary

If G is a graph in H′r , and T is the tree obtained by shrinking eachBr in G to one vertex, then every parity subgraph of G contains T .






If G is in H′r , then p(G ) =(2r2+3r−1)n−2(r+1)

4r2+4r−2− 1, which reduces to

2n−53 for cubic graphs.









Proof. Let T be the tree obtained by shrinking all the balloons inG . By the previous lemma, a parity subgraph must use all theedges in T .









Proof. Let T be the tree obtained by shrinking all the balloons inG . By the previous lemma, a parity subgraph must use all theedges in T . A parity subgraph of G must add r + 1 more edges ineach balloon (since Br has 2r + 3 vertices).









Proof. Let T be the tree obtained by shrinking all the balloons inG . By the previous lemma, a parity subgraph must use all theedges in T . A parity subgraph of G must add r + 1 more edges ineach balloon (since Br has 2r + 3 vertices). Hence,

p(G ) = c(G ) + (r + 1)b(G ) = r(n−2)−2pr

− 1 + (r + 1) (2r−1)n+22pr

= 2r(n−2)−4+(r+1)(2r−1)n+22pr − 1 =(2r2+3r−1)n−2(r+1)

4r2+4r−2− 1.




Definitions and Remarks for the upper bound

◮ An r -graph is an r -regular multigraph G on an even number ofvertices with the property that every edge-cut which separatesV (G ) into two sets of odd cardinality has size at least r .






◮ Note that if G is a 2-edge-connected cubic multigraph, thenG is 3-graph.







◮ More generally, if G is an (r − 1)-edge-connected r -regularmultigraph with even order, then G is r -graph.







◮ More generally, if G is an (r − 1)-edge-connected r -regularmultigraph with even order, then G is r -graph.

◮ Every r -edge-colorable r -regular graph is an r -graph.




Application of Edmonds’ Theorem

Lemma (Edmonds 1965)

If G is an r -graph, then there is an integer p and a family M ofperfect matchings such that each edge of G is contained in preciselyp members of M. (The members of M need not be distinct.)




Application of Edmonds’ Theorem

Lemma (Edmonds 1965)

If G is an r -graph, then there is an integer p and a family M ofperfect matchings such that each edge of G is contained in preciselyp members of M. (The members of M need not be distinct.)

Lemma (O and West 2009+)

Let G be a 2r -edge-connected (2r + 1)-regular multigraph. If G isedge-weighted, then there exists a perfect matching with weight atmost 12r+1W , where W =

∑

e∈E(G) we and we is the weight on an

edge e. For cubic graphs, the bound reduces to 13W .




Proof of the theorem

Proof. By Edmonds’ Lemma, |M|n2 =(2r+1)n

2 p, which implies that|M| = p(2r + 1).







Let M = {M1, ...,Mp(2r+1)}.







Let M = {M1, ...,Mp(2r+1)}.

Since∑

wMi = p∑

e∈E(G) we = pW where wMi is the total weightof all edges in Mi , the pigeonhole principle implies that a matchingMj with the smallest weight in the family has weight at most

12r+1W .




Chinese Postman Problem in Cubic graphs

Fn: the family of connected cubic graphs with n vertices.







If G is in Fn and n ≥ 10, then p(G ) ≤2n−5

3 ,with equality if G ∈ H′1.

Proof. If G is in Fn and has no balloons or n = 10, then G has aperfect matching and p(G ) = n/2 ≤ 2n−53 .










Therefore, we may assume that G has a balloon and n > 10.











Proceed by induction on n. Let e be a cut-edge.Let G1 and G2 be the components of G − e.











Proceed by induction on n. Let e be a cut-edge.Let G1 and G2 be the components of G − e.

Let G ′1 and G′

2 be the graphs obtained from G by replacing G2 andG1, respectively, with B1.




Chinese Postman Problem in Cubic Graphs

Every parity subgraph of G ′i contains e and and uses at least twoedges in B1. Hence, p(G

′

i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G

′

2) − 5.






′

i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G

′

2) − 5.

If neither G1 nor G2 is B1, then G′

1 and G′

2 are smaller than G .






′

i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G

′

2) − 5.


1 and G′


Letting ni = |V (G′

i )|, we have n = n1 + n2 − 10.By applying the induction hypothesis to both G ′1 and G

′

2,






′

i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G

′

2) − 5.


1 and G′


Letting ni = |V (G′

i )|, we have n = n1 + n2 − 10.By applying the induction hypothesis to both G ′1 and G

′

2,

p(G ) = p(G ′1) + p(G′

2) − 5 ≤2n1 − 5

3+

2n2 − 5

3− 5 =

2n − 5

3





1 1 1 3

Last case : every cut-edge is incident to a copy of B1.





1 1 1 3

Last case : every cut-edge is incident to a copy of B1.Let each edge have weight 1. Form G ′ by deleting all the balloons.





1 1 1 3

Last case : every cut-edge is incident to a copy of B1.Let each edge have weight 1. Form G ′ by deleting all the balloons.In G ′, replace each ′′thread ′′ through vertices of degree 2 with asingle edge whose weight is the length of the thread.





1 1 1 3

Last case : every cut-edge is incident to a copy of B1.Let each edge have weight 1. Form G ′ by deleting all the balloons.In G ′, replace each ′′thread ′′ through vertices of degree 2 with asingle edge whose weight is the length of the thread.The resulting weighted graph G ′′ has a perfect matching with atmost 1/3 of its total weight(=m−8b3 ).(Special case : when G ′ is a cycle, G has a perfect matching.)





1 1 1 3

Last case : every cut-edge is incident to a copy of B1.Let each edge have weight 1. Form G ′ by deleting all the balloons.In G ′, replace each ′′thread ′′ through vertices of degree 2 with asingle edge whose weight is the length of the thread.The resulting weighted graph G ′′ has a perfect matching with atmost 1/3 of its total weight(=m−8b3 ).(Special case : when G ′ is a cycle, G has a perfect matching.)Now, p(G ) ≤ p(G ′) + 3b ≤ m−8b3 + 3b =

3n−16b6 + 3b ≤

2n−53 .


Motivation and QuestionsPrevious Results using BalloonsSmallest Matching Number and Edge-ConnectivityChinese Postman Problem

Documents

Generalized Balloons and Chinese Postman Problems in ...suilo2/preprint/slide...Chinese Postman Problem Motivation and Questions In 1891, Petersen proved that every cubic graph without