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 $χ_{ON}(G)$. The decision problem that asks whether $χ_{ON}(G) \leq k$ is NP-complete. We obtain the following results: * Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound $χ_{ON}(G)\leq {\sf fvs}(G)+3$, where ${\sf fvs}(G)$ denotes the size of a minimum feedback vertex set of $G$. We show the improved bound of $χ_{ON}(G)\leq {\sf fvs}(G)+2$, which is tight, thereby answering an open question in the above paper. * We study the relation between $χ_{ON}(G)$ and the pathwidth of the graph $G$, denoted ${\sf pw}(G)$. The above paper from WADS 2019 showed the upper bound $χ_{ON}(G) \leq 2{\sf tw}(G)+1$ where ${\sf tw}(G)$ stands for the treewidth of $G$. This implies an upper bound of $χ_{ON}(G) \leq 2{\sf pw}(G)+1$. We show an improved bound of $χ_{ON}(G) \leq \lfloor \frac{5}{3}({\sf pw}(G)+1) \rfloor$. * We prove new bounds for $χ_{ON}(G)$ 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 $χ^*_{ON}(G)$ denote the minimum number of colors required to color $G$ as per this variant. Abel et. al. [SIDMA 2018] showed that $χ^*_{ON}(G) \leq 8$ when $G$ is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that $χ^*_{ON}(G) \leq 5$ for all planar $G$. All our bounds are a result of constructive algorithmic procedures.