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List-coloring in claw-free perfect graphs Lucas Pastor Joint-work with Sylvain Gravier and Frdric Maffray JGA 2015 04/11/15 Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 1 / 21 Coloring Coloring Given a

  1. 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

  2. Coloring 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

  3. Coloring 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

  4. List-coloring 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

  5. List-coloring 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

  6. List-coloring 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

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

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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. Perfect graph 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

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

  22. Claw-free perfect graph Claw-free graph The claw is the graph K 1 , 3 . A graph is said to be claw-free if it has no induced subgraph isomorphic to K 1 , 3 . Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 8 / 21

  23. Claw-free perfect graph Claw-free graph The claw is the graph K 1 , 3 . A graph is said to be claw-free if it has no induced subgraph isomorphic to K 1 , 3 . Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 8 / 21

  24. 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

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

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

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

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

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

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

  31. Peculiar graph Q 3 B 1 A 2 A 1 B 2 B 3 A 3 Q 2 Q 1 clique at least one non-edge complete adjacency Lucas Pastor (JGA 2015) List-coloring in claw-free perfect graphs 04/11/15 11 / 21

  32. 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

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

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

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

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

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

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

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

  40. 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

  41. 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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