Computational and Combinatorial Results on Conflict-free Choosability

2026-05-11

AI Summary

This study investigates the computational complexity and upper bounds of conflict-free closed- and open-neighborhood list coloring, denoted as CFCN* and CFON* choice numbers. Focusing on $K_{1,k}$-free graphs, the authors extend existing results for CFCN* coloring to the list coloring setting through a combination of combinatorial graph-theoretic analysis, probabilistic methods, and complexity theory, establishing the first upper bound of $O(k \ln \Delta)$ on the CFCN* choice number. Furthermore, they prove for the first time that deciding whether the CFCN* or CFON* choice number is at most 2 is NP-hard, even when $k = 1$ or $k = 2$, thereby revealing the inherent computational intractability of these problems.

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. Dębski and Przybyło [Journal of Graph Theory, 2022] showed that for any graph $G$ with maximum degree $Δ$, the CFCN$^*$ chromatic number of its line graph is $O(\ln Δ)$. 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 $O(k\ln Δ)$ 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 $O(k\ln Δ)$. 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 $k=1,2$.

Problem

• conflict-free choosability
• CFCN*
• CFON*
• list coloring
• computational hardness

Innovation

• conflict-free choosability
• list coloring
• K_{1,k}-free graphs
• NP-hardness
• neighborhood coloring