Connected greedy coloring of [formula omitted]-free graphs

Abstract

A connected ordering(v1,v2,…,vn) of V(G) is an ordering of the vertices such that vi has at least one neighbor in {v1,…,vi−1} for every i∈{2,…,n}. A connected greedy coloring (CGC for short) is a coloring obtained by applying the greedy algorithm to a connected ordering. This has been first introduced in 1989 by Hertz and de Werra, but still very little is known about this problem. An interesting aspect is that, contrary to the traditional greedy coloring, it is not always true that a graph has a connected ordering that produces an optimal coloring; this motivates the definition of the connected chromatic number ofG, which is the smallest value χc(G) such that there exists a CGC of G with χc(G) colors. An even more interesting fact is that χc(G)≤χ(G)+1 for every graph G (Benevides et al. 2014).In this paper, in the light of the dichotomy for the coloring problem restricted to H-free graphs given by Král’ et al. in 2001, we are interested in investigating the problems of, given an H-free graph G: (1). deciding whether χc(G)=χ(G); and (2). given also a positive integer k, deciding whether χc(G)≤k. We denote by Pt the path on t vertices, and by Pt+K1 the union of Pt and a single vertex. We have proved that Problem (2) has the same dichotomy as the coloring problem (namely, it is polynomial when H is an induced subgraph of P4 or of P3+K1, and it is NP-complete otherwise). As for Problem (1), we have proved that χc(G)=χ(G) always hold when H is an induced subgraph of P5 or of P4+K1, and that it is NP-complete to decide whether χc(G)=χ(G) when H is not a linear forest or contains an induced P9. We mention that some of the results involve fixed k and fixed χ(G).