List packing of graphs with bounded tree-width

TL;DR

This paper investigates list packing numbers of graphs with bounded tree-width, establishing bounds and exact values for specific cases, and shows that related decision problems are solvable efficiently.

Contribution

It improves bounds on the maximum packing number for graphs with bounded tree-width and proves linear-time solvability of related decision problems.

Findings

Proves that for d ≥ 3, t(d) ≤ 2d - 1.

Establishes that for d ≥ 2, t(d) ≥ d + 2.

Shows that determining if χ_l^{ extasteriskcentered}(G) ≤ k is solvable in linear time for fixed d, k.

Abstract

Assume $L$ is a $k$-assignment of a graph $G$. An $L$-packing $\phi$ of $G$ is a sequence $\phi=(\phi_1, \ldots, \phi_k)$ of $k$-mappings such that each $\phi_i$ is an $L$-coloring of $G$, and for each vertex $v$ of $G$, $\{\phi_1(v), \ldots, \phi_k(v)\} = L(v)$ (and hence $\phi_i(v) \ne \phi_j(v)$ when $i \ne j$). We say $G$ is list $k$-packable if for any $k$-assignment $L$ of $G$, there is an $L$-packing of $G$. The list packing number $\chi_l^{\star}(G)$ of $G$ is the minimum integer $k$ such that $G$ is $k$-packable. For a positive integer $d$, let $t(d)$ be the maximum packing number of graphs of tree-width at most $d$. It was known that $d+1 \le t(d) \le 2d$ for any $d$. In this paper, we prove that $t(d) \le 2d-1$ for $d \ge 3$, and $t(d) \ge d+2$ for $d \ge 2$. In particular, $t(2)=4$ and $t(3)=5$. Furthermore, we show that for constant positive integers $k, d$, the problem of determining $\chi_l^{\star}(G)\leq k$ or not for a graph $G$ of tree-width at most $d$ is solvable in linear time.