[PPT] - List-coloring in claw-free perfect graphs Lucas Pastor Joint-work PowerPoint Presentation

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List-coloring in claw-free perfect graphs

Lucas Pastor Joint-work with Sylvain Gravier and Frédéric Maffray

JGA 2015

04/11/15

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 1 / 21

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Coloring

Given a graph G, a (proper) k-coloring of the vertices of G is a mapping c : V (G) → {1, 2, . . . , k} for which every pair of adjacent vertices x, y satisfies c(x) = c(y).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 2 / 21

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Coloring

Given a graph G, a (proper) k-coloring of the vertices of G is a mapping c : V (G) → {1, 2, . . . , k} for which every pair of adjacent vertices x, y satisfies c(x) = c(y).

Chromatic number

The chromatic number of G, denoted by χ(G), is the smallest integer k such that G admits a k-coloring.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 2 / 21

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List-coloring

Let G be a graph. Every vertex v ∈ V (G) has a list L(v) of prescribed colors, we want to find a proper vertex-coloring c such that c(v) ∈ L(v).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 3 / 21

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List-coloring

Let G be a graph. Every vertex v ∈ V (G) has a list L(v) of prescribed colors, we want to find a proper vertex-coloring c such that c(v) ∈ L(v). When such a coloring exists, G is L-colorable.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 3 / 21

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List-coloring

Let G be a graph. Every vertex v ∈ V (G) has a list L(v) of prescribed colors, we want to find a proper vertex-coloring c such that c(v) ∈ L(v). When such a coloring exists, G is L-colorable.

Choice number

The choice number ch(G) of a graph G is the smallest k such that for every list assignment L of size k, the graph G is L-colorable.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 3 / 21

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List-coloring Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 4 / 21

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List-coloring Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 4 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Chromatic inequality

We have χ(G) ≤ ch(G) for every graph G. There are graphs for which χ(G) = ch(G) (in fact, the gap can be arbitrarily large).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 5 / 21

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List-coloring

Vizing’s conjecture

For every graph G, χ(L(G)) = ch(L(G)). In other words, χ′(G) = ch′(G) with ch′(G) the list chromatic index of G.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 6 / 21

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List-coloring

Vizing’s conjecture

For every graph G, χ(L(G)) = ch(L(G)). In other words, χ′(G) = ch′(G) with ch′(G) the list chromatic index of G.

Conjecture [Gravier and Maffray, 1997]

For every claw-free graph G, χ(G) = ch(G).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 6 / 21

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List-coloring

Vizing’s conjecture

For every graph G, χ(L(G)) = ch(L(G)). In other words, χ′(G) = ch′(G) with ch′(G) the list chromatic index of G.

Conjecture [Gravier and Maffray, 1997]

For every claw-free graph G, χ(G) = ch(G).

Special case

We are interested in the case where G is perfect.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 6 / 21

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List-coloring

Vizing’s conjecture

For every graph G, χ(L(G)) = ch(L(G)). In other words, χ′(G) = ch′(G) with ch′(G) the list chromatic index of G.

Conjecture [Gravier and Maffray, 1997]

For every claw-free graph G, χ(G) = ch(G).

Special case

We are interested in the case where G is perfect.

Theorem [Gravier, Maffray, P.]

Let G be a claw-free perfect graph with ω(G) ≤ 4. Then χ(G) = ch(G).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 6 / 21

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Perfect graph

A graph G is called perfect if for every induced subgraph H of G, χ(H) = ω(H).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 7 / 21

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Perfect graph

A graph G is called perfect if for every induced subgraph H of G, χ(H) = ω(H).

Strong Perfect Graph Theorem

A graph G is perfect if and only if G does not contain an odd hole nor an

dd antihole.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 7 / 21

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Claw-free perfect graph

Claw-free graph

The claw is the graph K1,3. A graph is said to be claw-free if it has no induced subgraph isomorphic to K1,3.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 8 / 21

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Claw-free perfect graph

Claw-free graph

The claw is the graph K1,3. A graph is said to be claw-free if it has no induced subgraph isomorphic to K1,3.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 8 / 21

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Claw-free perfect graph

Theorem [Chvátal and Sbihi, 1988]

Every claw-free perfect graph either has a clique-cutset, or is a peculiar graph, or is an elementary graph.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 9 / 21

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Claw-free perfect graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 10 / 21

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Claw-free perfect graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 10 / 21

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Claw-free perfect graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 10 / 21

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Claw-free perfect graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 10 / 21

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Claw-free perfect graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 10 / 21

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Claw-free perfect graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 10 / 21

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Peculiar graph

A3 A2 A1 B1 B3 B2 Q1 Q2 Q3 clique at least one non-edge complete adjacency

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 11 / 21

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Elementary graph

Theorem [Maffray and Reed, 1999]

A graph G is elementary if and only if it is an augmentation of the line-graph H (called the skeleton of G) of a bipartite multigraph B (called the root graph of G).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 12 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Elementary graph Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 13 / 21

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Theorem and sketch of the proof

Theorem [Gravier, Maffray, P.]

Let G be a claw-free perfect graph with ω(G) ≤ 4. Then χ(G) = ch(G).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 14 / 21

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Theorem and sketch of the proof

Lemma

Let G be a connected claw-free perfect graph that contains a peculiar

subgraph. Then G is peculiar.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 15 / 21

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Theorem and sketch of the proof

Lemma

Let G be a connected claw-free perfect graph that contains a peculiar

subgraph. Then G is peculiar.

Lemma

Let G be a peculiar graph with ω(G) ≤ 4 (unique in this case). Then G is 4-choosable.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 15 / 21

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Theorem and sketch of the proof

Theorem [Galvin, 1994]

Let G be the line-graph of a bipartite multigraph. Then χ(G) = ch(G).

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 16 / 21

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Theorem and sketch of the proof

Theorem [Galvin, 1994]

Let G be the line-graph of a bipartite multigraph. Then χ(G) = ch(G).

Proof for the elementary graphs

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 16 / 21

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Theorem and sketch of the proof

Theorem [Galvin, 1994]

Let G be the line-graph of a bipartite multigraph. Then χ(G) = ch(G).

Proof for the elementary graphs

By induction on the number h of augmented flat edges

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 16 / 21

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Theorem and sketch of the proof

Theorem [Galvin, 1994]

Let G be the line-graph of a bipartite multigraph. Then χ(G) = ch(G).

Proof for the elementary graphs

By induction on the number h of augmented flat edges If h = 0, by Galvin’s theorem the base case is verified

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 16 / 21

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Theorem and sketch of the proof

Theorem [Galvin, 1994]

Let G be the line-graph of a bipartite multigraph. Then χ(G) = ch(G).

Proof for the elementary graphs

By induction on the number h of augmented flat edges If h = 0, by Galvin’s theorem the base case is verified We show that we can always extend the coloring to the last augmented flat edge

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 16 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 17 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 17 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 17 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 17 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 17 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 17 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2]

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2] Suppose that G1 is colored and that G2 is an elementary graph

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2] Suppose that G1 is colored and that G2 is an elementary graph We want to extend this coloring to G2

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2] Suppose that G1 is colored and that G2 is an elementary graph We want to extend this coloring to G2

Proof of the main theorem

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2] Suppose that G1 is colored and that G2 is an elementary graph We want to extend this coloring to G2

Proof of the main theorem

The colors on C are forced by the coloring of G1

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2] Suppose that G1 is colored and that G2 is an elementary graph We want to extend this coloring to G2

Proof of the main theorem

The colors on C are forced by the coloring of G1 This is equivalent to reducing the list size on the vertices of C

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof

Hypothesis of the main theorem

Let G be a claw-free perfect graph and C a clique cutset. The graph G \ C has a two disjoint set of vertices A1 and A2 Let G1 = G[C ∪ A1] and G2 = G[C ∪ A2] Suppose that G1 is colored and that G2 is an elementary graph We want to extend this coloring to G2

Proof of the main theorem

The colors on C are forced by the coloring of G1 This is equivalent to reducing the list size on the vertices of C Thanks to a Galvin’s argument, we can show that G2 is list-colorable with restriction of the list size of C

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 18 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 19 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 19 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 19 / 21

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Theorem and sketch of the proof Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 19 / 21

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Conclusion and perspectives

Perspectives

Prove it for the general case! Or disprove it?!

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 20 / 21

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Conclusion and perspectives

Perspectives

Prove it for the general case! Or disprove it?!

A word on our method

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 20 / 21

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Conclusion and perspectives

Perspectives

Prove it for the general case! Or disprove it?!

A word on our method

Proving that elementary graphs are chromatic-choosable by induction

n the number of augmented flat edges gives us interesting tools for

the extension of a coloring to an elementary graph.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 20 / 21

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Conclusion and perspectives

Perspectives

Prove it for the general case! Or disprove it?!

A word on our method

Proving that elementary graphs are chromatic-choosable by induction

n the number of augmented flat edges gives us interesting tools for

the extension of a coloring to an elementary graph. It seems to be hard to use this trick for the general case.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 20 / 21

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Conclusion and perspectives

Perspectives

Prove it for the general case! Or disprove it?!

A word on our method

Proving that elementary graphs are chromatic-choosable by induction

n the number of augmented flat edges gives us interesting tools for

the extension of a coloring to an elementary graph. It seems to be hard to use this trick for the general case. We tried Galvin like arguments without any success.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 20 / 21

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Conclusion and perspectives

Perspectives

Prove it for the general case! Or disprove it?!

A word on our method

Proving that elementary graphs are chromatic-choosable by induction

n the number of augmented flat edges gives us interesting tools for

the extension of a coloring to an elementary graph. It seems to be hard to use this trick for the general case. We tried Galvin like arguments without any success. What about peculiar graphs?

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 20 / 21

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Conclusion and perspectives

Thank you for listening.

Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 21 / 21