On the b-Chromatic Number of Graphs

Abstract

Abstract. The b-chromatic number b(G) of a graph G = (V, E) is the largest integer k such that G admits a vertex partition into k independent sets Xi (i = 1, . . . , k) such that each Xi contains a vertex xi adjacent to at least one vertex of each Xj, j = i. We discuss on the tightness of some bounds on b(G) and on the complexity of determining b(G). We also determine the asymptotic behavior of b(Gn,p) for the random graph, within the accuracy of a multiplicative factor 2 + o(1) as n → ∞.

A new Mathematical Programming Model for the Green Vehicle Routing Problem

“…But since the coloring problem is NP-complete, χ(G) cannot always be reached. Irving and Manlove [10], interested in the worst case scenario, defined a b-coloring as a coloring of G that has at least one b-vertex in each of its color classes, and the b-chromatic number of G as the maximum number of colors b (G) used by a b-coloring of G. Finding b(G) is NP-complete [10], even if G is bipartite [13], chordal [9], or a line graph [5].…”

“…It is known that K n,n minus a perfect matching only admits b-colorings with 2 and n colors, for n ∈ N [13]. Also, for every finite S ⊂ N − {1}, there exists a graph G that admits a b-coloring with k colors iff k ∈ S [2].…”

A Characterization of b‐Perfect Graphs

“…One of h, j is not equal to 0, say j = 0. So A j = ∅ and, by (20), x has no neighbor in that set; but then {a j , u g , u h , x, v i , v j } induces an F 10 . Now we may assume that the four integers g, h, i, j are different.…”

“…Now, by (20), x has a neighbor in A j , or in B h , but not in both. If x has a neighbor in B h , then {a j , u h , u g , x, v h , v j } induces an F 10 . If x has a neighbor in A j , then {b h , v j , v i , x, u h , u j } induces an F 10 .…”

On b-vertex and b-edge critical graphs

“…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]).…”