The b-chromatic number of a graph

Abstract

The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V 1 , V 2 ,. .. , V k into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the b-chromatic number, ϕ(G), of G. We prove that determining ϕ(G) is NP-hard for general graphs, but polynomial-time solvable for trees.

On b-vertex and b-edge critical graphs

“…Irving and Manlove [14,21] define the m-degree m(G) of G to be the largest integer t such that G has at least t vertices of degree at least t − 1. Thus every graph G satisfies the following.…”

“…Irving and D.F. Manlove ( [14,21]). They proved that determining b(G) is N P -hard for general graphs, even when it is restricted to the class of bipartite graphs ( [20]), but it is polynomial for trees ( [14,21]).…”

“…Manlove ( [14,21]). They proved that determining b(G) is N P -hard for general graphs, even when it is restricted to the class of bipartite graphs ( [20]), but it is polynomial for trees ( [14,21]). The N P -completeness results have incited researchers to establish bounds on the b-chromatic number in general or to find its exact values for subclasses of graphs (see [2, 3, 6-8, 10, 12, 15, 18-20, 22, 23]).…”

A Characterization of b‐Perfect Graphs

“…It follows that u i ∈ K. Then b h is not adjacent to u i , for otherwise {a 3 , a 5 , b h , u i , u h , y, y ′ } induces an F 11 . Vertex b h must have a neighbor v i of color i.…”

On vertex b-critical trees

“…We call any such vertex a b-vertex. The concept of b-coloring was introduced by Irving and Manlove [3,4]. The b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors.…”