Homogeneous sets, clique-separators, critical graphs, and optimal χ-binding functions

Abstract

Given a set H of graphs, let f H ⋆ : N > 0 → N > 0 be the optimal χ -binding function of the class of H -free graphs, that is, f H ⋆ ( ω ) = max { χ ( G ) : G is H -free, ω ( G ) = ω } . In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal χ -binding functions for subclasses of P 5 -free graphs and of ( C 5 , C 7 , … ) -free graphs. In particular, we prove the following for each ω ≥ 1 : (i) f { P 5 , b a n n e r } ⋆ ( ω ) = f { 3 K 1 } ⋆ ( ω ) ∈ Θ ( ω 2 / log ( ω ) ) , (ii) f { P 5 , c o - b a n n e r } ⋆ ( ω ) = f { 2 K 2 } ⋆ ( ω ) ∈ O ( ω 2 ) , (iii) f { C 5 , C 7 , … , b a n n e r } ⋆ ( ω ) = f { C 5 , 3 K 1 } ⋆ ( ω ) ∉ O ( ω ) , and. (iv) f { P 5 , C 4 } ⋆ ( ω ) = ⌈ ( 5 ω − 1 ) / 4 ⌉ . We also characterise, for each of our considered graph classes, all graphs G with χ ( G ) > χ ( G − u ) for each u ∈ V ( G ) . From these structural results, Reed’s conjecture – relating chromatic number, clique number, and maximum degree of a graph – follows for ( P 5 , b a n n e r ) -free graphs.