On minimally b-imperfect graphs

Abstract

a b s t r a c tA b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.

A Characterization of b‐Perfect Graphs

“…There must be an edge xz with x ∈ X 1 ∪· · ·∪X 6 and z ∈ Z, for otherwise X 1 ∪· · ·∪X 6 is a homogeneous set, which contradicts Lemma 2. 9. By item 10 of Lemma 3.1, x is in X ′′ 1 ∪X ′′ 4 ∪X 6 .…”

“…, B q is a homogeneous set, so it satisfies the properties described in Lemma 2. 9. It follows (recall Lemma 2.3) that: …”

“…Theorem 1.3 (Hoàng, Linhares Sales, Maffray [9]) Conjecture 1 holds for 3-colorable graphs and for diamond-free graphs. [12]) Conjecture 1 holds for chordal graphs.…”

“…Hoàng and Kouider [8] proved the b-perfectness of some classes of graphs, and asked for a good characterization of the whole class of b-perfect graphs. Hoàng, Linhares Sales and Maffray [9] proposed the conjecture below. Here we solve the problem by establishing the validity of the conjecture.…”

“…, F 22 } be the set of graphs depicted in Figure 1. [9]) A graph is b-perfect if and only if it is F-free.…”