Graph Saturation in Colorelyse/Colloquium_UAA15.pdf · Ferrara-Kim-Yeager (UCD, UIUC) Graph...

Graph Saturation in Color

Michael Ferrara Jaehoon Kim Elyse Yeager*

yeager2@illinois.edu

UAA Mathematics ColloquiumFebruary 2015

Ferrara-Kim-Yeager (UCD, UIUC) Graph Saturation in Color Feb 2015 1 / 22

Outline

1 Introduction to Graphs

2 Graph Saturation

3 Ramsey Theory

4 Colored Graph Saturation

What Do You Know About Graphs?

Graph Basics

vertex

|G | : number of vertices in G

‖G‖ : number of edges in G

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Graph Basics

vertex

Subgraph:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Common Graphs

Matching:

Clique:

Outline

2 Graph Saturation

3 Ramsey Theory

Avoid a forbidden subgraph, but add as many edges as possible.

Example:

Let the triangle be forbidden.

Example:

This graph is triangle saturated

Example:

Definition

A graph G is H-saturated if:

1 H is not a subgraph of G and

2 If we add any edge to G , H is a subgraph of the resulting graph.

Example:

Definition

1 H is not a subgraph of G

Example:

Definition

1 H is not a subgraph of G and

Saturation Number

7 edges 8 edges 5 edges

Definition: (Erdos-Hajnal-Moon)

The saturation number of a graph H and a number n is the minimumnumber of edges in a graph on n vertices that is H saturated.

sat(n;H) := min{‖G‖ : |G | = n and G is H saturated}

Given the above examples:

sat(6; triangle) ≤ 5

Saturation Number

7 edges

8 edges 5 edges

Saturation Number

7 edges 8 edges

5 edges

Saturation Number

sat(6; triangle)

Saturation Number

Outline

2 Graph Saturation

3 Ramsey Theory

Edge Coloring

Definition:

An edge coloring of a graph is an assignment of a color to each edge.

Definition:

A graph is monochromatic if all its edges are assigned the same color.

Edge Coloring

Definition:

Colors: red, blue.Colors: red, blue.Colors: red, blue.

Definition:

Edge Coloring

Definition:

Colors: red, blue.

Colors: red, blue.Colors: red, blue.

Definition:

Edge Coloring

Definition:

Colors: red, blue.

Definition:

Edge Coloring

Definition:

Colors: red, blue.Colors: red, blue.

Colors: red, blue.

Definition:

Edge Coloring

Definition:

Ramsey Problems

An edge-coloring with no forbidden monochromatic subgraph.

Example:

Color red and blue; forbid red triangle and blue triangle.

good coloring bad coloring bad coloring

Ramsey Problems

Example:

Ramsey Problems

Example:

Ramsey Problems

Example:

good coloring

bad coloring bad coloring

Ramsey Problems

Example:

good coloring bad coloring

bad coloring

Ramsey Problems

Example:

Ramsey Problems

Sometimes a good coloring doesn’t exist!

Ramsey Problems

Example: Any red-blue coloring of the edges of K6 contains amonochromatic red triangle or a monochromatic blue triangle.

no good coloring exists

Ramsey Problems

Ramsey’s Theorem

Given any collection of forbidden subgraphs, and any sufficiently largeclique, no good coloring exists.

The lowest “sufficiently large” number for a given collection of subgraphsis called the Ramsey Number.

R(3, 3) = 6 (two triangles)

R(4, 4) = 18

43 ≤ R(5, 5) ≤ 49

Ramsey Problems

Ramsey’s Theorem

R(4, 4) = 18

43 ≤ R(5, 5) ≤ 49

Ramsey Problems

Ramsey’s Theorem

R(4, 4) = 18

43 ≤ R(5, 5) ≤ 49

Ramsey Problems

Ramsey’s Theorem

R(4, 4) = 18

43 ≤ R(5, 5) ≤ 49

Ramsey Problems

Ramsey’s Theorem

Erdos-Szekeres: Given N, any sufficiently large collection of points ingeneral position contains a subset forming the vertices of a convex N-gon

Van der Waerden: Any coloring of the natural numbers containsarbitrarily long monochromatic arithmetic sequences.

Green-Tao: The sequence of prime numbers contains arbitrarily longarithmetic progressions.

Outline

2 Graph Saturation

3 Ramsey Theory

Marrying The Two

Recall:

A graph G is H-saturated if G contains no H subgraph, but adding anyedge to G creates an H subgraph.

Ramsey version of saturation:

Given forbidden graphs H1, . . . ,Hk (in colors 1, . . . , k respectively), we saya graph G is (H1, . . . ,Hk)-saturated if a good edge-coloring of G exists(using colors 1, . . . , k), but if we add any edge to G , all colorings are bad.

Marrying The Two

Recall:

A graph G is H-saturated if G contains no H subgraph, but adding anyedge to G creates an H subgraph.

Ramsey version of saturation:

Given forbidden graphs H1, . . . ,Hk (in colors 1, . . . , k respectively), we saya graph G is (H1, . . . ,Hk)-saturated if a good edge-coloring of G exists(using colors 1, . . . , k), but if we add any edge to G , all colorings are bad.

Marrying The Two

Example: the graph below is

-saturated.

First: show a good coloring exists.

Second: show no good coloring exists if we add any edge.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

Marrying The Two

-saturated.

rsat(n;H1, . . . ,Hk)

Again, we are interested in (H1, . . . ,Hk)-saturated graphs with as fewedges as possible.

Definition:

For a number n and forbidden subgraphs H1, . . . ,Hk , we define

rsat(n;H1, . . . ,Hk)

to be the minimum number of edges over all n-vertex graphs that are(H1, . . . ,Hk)-saturated.

Previous Example:

-saturated, so rsat

≤ 6.

rsat(n;H1, . . . ,Hk)

Definition:

rsat(n;H1, . . . ,Hk)

Previous Example:

-saturated

, so rsat

≤ 6.

rsat(n;H1, . . . ,Hk)

Definition:

rsat(n;H1, . . . ,Hk)

Previous Example:

-saturated, so rsat

≤ 6.

Forbidding Cliques: Ramsey Numbers

Definition:

Let R = R(c1, . . . , ct) be the smallest natural number so that, forforbidden cliques Kc1 , . . . ,Kct , no good coloring of KR exists.

Example: R(3, 3) = 6

no good coloring of K6 exists a good coloring of K5 exists

Definition:

no good coloring of K6 exists

a good coloring of K5 exists

Definition:

Forbidding Cliques: Hanson-Toft

Let R = R(c1, . . . , ct). The construction below is (Kc1 , . . . ,Kct )-saturated.

KR−2 n − (R − 2)

Hanson-Toft

Conjecture: The construction above has the fewest possible edges.

KR−2 n − (R − 2)

Hanson-Toft

KR−2 n − (R − 2)

A good coloring exists

If we add any edge, NO good coloring exists

Hanson-Toft

KR−2 n − (R − 2)

Hanson-Toft

KR−2 n − (R − 2)

Hanson-Toft

KR−2 n − (R − 2)

Hanson-Toft

KR−2 n − (R − 2)

A good coloring existsIf we add any edge, NO good coloring exists

Hanson-Toft

KR−2 n − (R − 2)

Hanson-Toft

KR−2 n − (R − 2)

Hanson-Toft

KR−2 n − (R − 2)

Hanson-Toft

Hanson-Toft Conjecture

sat(n;Rmin(Kk1 , . . . ,Kkt )) =

)n < r(r−2

)+ (r − 2)(n − r + 2) n ≥ r

Chen, Ferrara, Gould, Magnant, Schmitt; 2011

sat(n;Rmin(K3,K3)) =

)n < 6 = r

4n − 10 n ≥ 56

Hanson-Toft

)n < r(r−2

)+ (r − 2)(n − r + 2) n ≥ r

)n < 6 = r

4n − 10 n ≥ 56

Hanson-Toft

)n < r(r−2

)+ (r − 2)(n − r + 2) n ≥ r

)n < 6 = r

4n − 10 n ≥ 56

Matching Saturation: Mader, 1973

Forbidden:

Graph:

No forbidden subgraph

Adding any edge createsforbidden subgraph

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

No forbidden subgraph Adding any edge createsforbidden subgraph

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidden:

Graph:

Forbidding Matchings: Ferrara-Kim-Y

Example: Forbidden matching on 10 edges.

Colored Example: Forbidden

Iterated Recoloring: Ferrara-Kim-Yeager

Use results from (uncolored) saturation in the Ramsey version.

Example: Forbidden graphs

good coloring

make red-heavy

take red subgraph

This (uncolored!) subgraph is saturated.

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

good coloring

make red-heavy

take red subgraph

Thanks for Listening!

Graph Saturation in Colorelyse/Colloquium_UAA15.pdf · Ferrara-Kim-Yeager (UCD, UIUC) Graph...

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