Conflict-Free Coloring Bounds 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 $\chi _{ON}(G)$. The decision problem that asks whether $\chi _{ON}(G)\le k$ is NP-complete. Structural as well as algorithmic aspects of this problem have been well studied. We obtain the following results for $\chi _{ON}(G)$:

• Bodlaender, Kolay and Pieterse (WADS 2019) showed the upper bound $\chi _{ON}(G)\le \mathsf{fvs}(G) + 3$, where $\mathsf{fvs}(G)$ denotes the size of a minimum feedback vertex set of G. We show the improved bound of $\chi _{ON}(G)\le \mathsf{fvs}(G) + 2$, which is tight, thereby answering an open question in the above paper.

• We study the relation between $\chi _{ON}(G)$ and the pathwidth of the graph G, denoted $\mathsf{pw}(G)$. The above paper from WADS 2019 showed the upper bound $\chi _{ON}(G)\le 2\mathsf{tw}(G) + 1$ where $\mathsf{tw}(G)$ stands for the treewidth of G. This implies an upper bound of $\chi _{ON}(G)\le 2\mathsf{pw}(G) + 1$. We show an improved bound of $\chi _{ON}(G)\le \left\lfloor \frac{5}{3}(\mathsf{pw}(G) + 1)\right\rfloor$.

• We prove new bounds for $\chi _{ON}(G)$ with respect to the structural parameters neighborhood diversity and distance to cluster, improving the existing results of Gargano and Rescigno (Theor. Comput. Sci. 2015) and Reddy (Theor. Comput. Sci. 2018), respectively. Furthermore, our techniques also yield improved bounds for the closed neighborhood variant of the problem.

• We prove bounds for $S_k$-free graphs where $S_k$ is a star on $k+1$ vertices. For a graph G with maximum degree $\Delta$, it is known that $\chi _{ON}(G)\le \Delta + 1$ and this bound is tight in general. When G is $S_k$-free, we show that $\chi _{ON}(G)= O(k\cdot \log ^{2+\epsilon }\Delta )$, for any $\epsilon > 0$. In particular, when G is claw-free, this implies that $\chi _{ON}(G)= O(\log ^{2 + \epsilon }\Delta )$. Further, we show existence of claw-free graphs that require $\Omega (\log \Delta )$ colors.

• We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let $\chi ^*_{ON}(G)$ denote the minimum number of colors required to color G as per this variant. Abel et al. (SIDMA 2018) showed that $\chi ^*_{ON}(G)\le 8$ when G is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that $\chi ^*_{ON}(G)\le 5$ for all planar G. This approach also yields the bound $\chi ^*_{ON}(G)\le 4$ for all outerplanar G.

All our bounds are a result of constructive algorithmic procedures.