On graphs without P5 and P5

Perfect coloring and linearly χ‐bound P6‐free graphs

“…In this paper, we derive decomposition theorems for {P 6 , K 1 + P 4 }-free graphs, {P 5 , K 1 + P 4 }-free graphs and {P 5 , K 1 + C 4 }-free graphs, and deduce linear χ-binding functions for these classes of graphs. Using the same techniques, we also obtain an optimal χ-binding function for {P 5 , C 4 }-free graphs which is an improvement over that given in [11].…”

“…While the linear χ-binding function of Theorem 16 is optimal, the linear χ-binding functions of Corollary 11 and Theorems 15, 18 do not seem to be optimal. Since the class of {P 5 , P c 5 }-free graphs admits no linear χ-binding function [11], it is not possible to replace K 1 + P 4 with K 1 + P n (n ≥ 5) in the hypothesis of Theorems 10 and 13. In this respect, we have following two problems.…”

“…We next obtain an improvement over Theorem B by a simple input to the arguments in the proof of Theorem A [11]. Moreover, our bound is optimal (see [8] for optimal graphs attaining the upper bounds).…”

“…It is known that [11], the class of P 5 -free graphs (in fact the smaller class of {P 5 , P c 5 }-free graphs) does not admit a linear χ-binding function.…”

“…While P 4 -free graphs have received much attention, P 5 -free graphs and P 6 -free graphs have received attention only recently [1,4,5,11,14]. The interest began when these graphs arose in connection with a study on graph grammars [10], and it came to be known that for P 5 -free graphs, the complexity of the maximum stable set problem is unknown [14] and that the P 4 -free coloring problem is NP-hard [13].…”