A Characterization of $(4,2)$-Choosable Graphs

TL;DR

This paper proves a conjecture that characterizes (4,2)-choosable graphs, advancing understanding of graph coloring with list assignments and specific intersection constraints.

Contribution

It provides a complete proof of Meng, Puleo, and Zhu's conjecture on the characterization of (4,2)-choosable graphs.

Findings

Confirmed the conjecture on (4,2)-choosable graphs

Established necessary and sufficient conditions for (4,2)-choosability

Enhanced understanding of list coloring with intersection constraints

Abstract

A graph $G$ is \emph{$(a,b)$-choosable} if given any list assignment $L$ with $|L(v)|=a$ for each $v\in V(G)$ there exists a function $\varphi$ such that $\varphi(v)\in L(v)$ and $|\varphi(v)|=b$ for all $v\in V(G)$, and whenever vertices $x$ and $y$ are adjacent $\varphi(x)\cap \varphi(y)=\emptyset$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture.