Choosability with Separation of Cycles and Outerplanar Graphs

TL;DR

This paper investigates the maximum separation number in list coloring with separation for cycles, cacti, and outerplanar graphs, providing exact values and bounds for these classes.

Contribution

It determines the separation and free-separation numbers for cycles and cacti, and establishes lower bounds for outerplanar graphs with girth at least 5.

Findings

Exact separation number for cycles.

Free-separation number for cacti.

Lower bounds for outerplanar graphs with girth ≥ 5.

Abstract

We consider the following list coloring with separation problem of graphs: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|\le a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$ for any vertex $v$ and $\varphi(u)\cap \varphi(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth $g\ge 5$.