Dynamic choosability of triangle-free graphs and sparse random graphs

TL;DR

This paper investigates the dynamic choosability of graphs, establishing bounds relating it to the choice number for various classes including bounded degree graphs, random graphs, and triangle-free regular graphs.

Contribution

It proves that the r-dynamic choosability is asymptotically close to the choice number for graphs with bounded degree ratios, and provides bounds for random and triangle-free regular graphs.

Findings

For graphs with bounded degree ratio, ${ m ch}_r(G)$ is asymptotically at most $(1+o(1)){ m ch}(G)$.

In random graphs $G(n,p)$ with $p$ between $2/n$ and 1/2, ${ m ch}_2(G)$ is at most ${ m ch}(G)+C$ asymptotically almost surely.

For triangle-free regular graphs, ${ m ch}_2(G)$ is at most ${ m ch}(G)+86$.

Abstract

The \textit{$r$-dynamic choosability} of a graph $G$, written ${\rm ch}_r(G)$, is the least $k$ such that whenever each vertex is assigned a list of at least $k$ colors a proper coloring can be chosen from the lists so that every vertex $v$ has at least $\min\{d_G(v),r\}$ neighbors of distinct colors. Let ${\rm ch}(G)$ denote the choice number of $G$. In this paper, we prove ${\rm ch}_r(G)\leq (1+o(1)){\rm ch}(G)$ when $\frac{\Delta(G)}{\delta(G)}$ is bounded. We also show that there exists a constant $C$ such that for the random graph $G=G(n,p)$ with $\frac{2}{n}<p\leq \frac{1}{2}$, it holds that ${\rm ch}_2(G)\leq {\rm ch}(G) + C$, asymptotically almost surely. Also if $G$ is triangle-free regualr graph, then ${\rm ch}_2(G)\leq {\rm ch}(G)+86$ holds.