Rogers Mathew

Shiwali Gupta, Rogers Mathew

A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the '(partial) conflict-free chromatic number' of $\mathcal{H}$. It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ with maximum degree $Δ$ is at most $O(\ln^2 Δ)$. In this paper, we extend the $O(\ln^2 Δ)$ upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers.

Shiwali Gupta, Rogers Mathew

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 $k=1,2$.