Computational and Combinatorial Results on Conflict-free Choosability

TL;DR

This paper studies conflict-free neighborhood colorings in graphs, extending known bounds to list variants and proving NP-hardness for computing certain coloring parameters.

Contribution

It generalizes existing bounds on conflict-free neighborhood choice numbers to list coloring and establishes NP-hardness results for these parameters.

Findings

CFCN* choice number of K_{1,k}-free graphs is O(k log Δ)

Extended bounds from line graphs to list coloring variants

Proved NP-hardness for deciding CFCN*/CFON* choice numbers for k=1,2

Abstract

The conflict-free closed neighborhood (CFCN$^*$) chromatic number of a graph $G = (V,E)$ is the smallest positive integer $k$ for which there exists a coloring of a subset of vertices using $k$ colors such that, for every vertex in $V$, there exists a color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) chromatic number is defined analogously. In this paper, we study `list variants' of the above-mentioned coloring parameters. The conflict-free closed neighborhood (CFCN$^*$) choice number of a graph $G = (V,E)$ is the smallest positive integer $k$ such that for every assignment of lists of size $k$ to its vertices, there exists a coloring of a subset of vertices, say $V'$, in which (i) every vertex in $V'$ receives a color from its list, and (ii) for every vertex in $V$ there exists some color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) choice number is defined analogously. D\k{e}bski and Przyby\l o [Journal of Graph Theory, 2022] showed that for any graph $G$ with maximum degree $\Delta$, the CFCN$^*$ chromatic number of its line graph is $O(\ln \Delta)$. This result was later extended to claw-free graphs by Bhyravarapu et al. [Journal of Graph Theory, 2025], who proved that every $K_{1,k}$-free graph $G$ admits a CFCN$^*$ coloring using $O(k\ln \Delta)$ colors. In this paper, we generalize this result to the list setting and show that every $K_{1,k}$-free graph $G$ has a CFCN$^*$ choice number of $O(k\ln \Delta)$. Further, we answer some questions concerning the hardness of computing CFCN$^*$/CFON$^*$ choice numbers posed by Gupta and Mathew [SOFSEM, 2026]; in particular, we show that it is NP-hard to determine whether the CFCN$^*$/CFON$^*$ choice number a graph is equal to $k$, for $k=1,2$.