Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods
Abstract
In an undirected graph G, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of G is the CFON chromatic number of G, denoted by ON(G). The decision problem that asks whether ON(G) ≤ k is NP-complete. We obtain the following results: * Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound ON(G)≤ fvs(G)+3, where fvs(G) denotes the size of a minimum feedback vertex set of G. We show the improved bound of ON(G)≤ fvs(G)+2, which is tight, thereby answering an open question in the above paper. * We study the relation between ON(G) and the pathwidth of the graph G, denoted pw(G). The above paper from WADS 2019 showed the upper bound ON(G) ≤ 2 tw(G)+1 where tw(G) stands for the treewidth of G. This implies an upper bound of ON(G) ≤ 2 pw(G)+1. We show an improved bound of ON(G) ≤ 53( pw(G)+1) . * We prove new bounds for ON(G) with respect to the structural parameters neighborhood diversity and distance to cluster, improving existing results. * We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let *ON(G) denote the minimum number of colors required to color G as per this variant. Abel et. al. [SIDMA 2018] showed that *ON(G) ≤ 8 when G is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that *ON(G) ≤ 5 for all planar G. All our bounds are a result of constructive algorithmic procedures.