Connected greedy coloring of H-free graphs

A connected ordering (v(1), v(2) ...., v(n)) of V(G) is an ordering of the vertices such that vi has at least one neighbor in {v(1), ..., v(i-1)} for every i is an element of {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 of G, which is the smallest value chi(c)(G) such that there exists a CGC of G with chi(c)(G) colors. An even more interesting fact is that chi(c)(G) <= chi(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 Kral' et al. in 2001, we are interested in investigating the problems of, given an H-free graph G: (1). deciding whether chi(c)(G) = chi(G); and (2). given also a positive integer k, deciding whether chi(c)(G) <= k. We denote by P-t the path on t vertices, and by P-t +K-1 the union of P-t 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 P-4 or of P-3 +K-1, and it is NP-complete otherwise). As for Problem (1), we have proved that chi(c)(G) = chi(G) always hold when H is an induced subgraph of P-5 or of P-4 +K-1, and that it is NP-complete to decide whether chi(c)(G) = chi(G) when H is not a linear forest or contains an induced P-9. We mention that some of the results involve fixed k and fixed chi(G). (C) 2020 Elsevier B.V. All rights reserved.