Equitable list coloring of sparse graphs

TL;DR

This paper investigates equitable list colorings of sparse graphs, introducing the stronger strongly equitable (SE) variant to tightly bound color-class sizes. It develops a potential-function framework with $\rho_{G}^{k}$ and shows that if $\rho_{G}^{k}\le 2-\sigma_{G}^{k}$, then $G$ is SE-$k$-choosable, thereby unifying the analysis for $k=3$ and $k=4$. The authors prove two sharp main results: every $(\frac{7}{6},\frac{1}{3})$-sparse graph with $\delta(G)\ge 2$ is equitably $3$-colorable and equitably $3$-choosable, and every $(\frac{5}{4},\frac{1}{2})$-sparse graph with $\delta(G)\ge 2$ is equitably $4$-colorable and equitably $4$-choosable; both bounds are tight. The methods combine safe-bug decompositions, a minimal counterexample analysis, and a discharging argument, yielding sharp results that extend and strengthen prior work on equitable coloring under sparse-graph constraints and suggest pathways to higher-$k$ extensions.

Abstract

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/k\rceil$ times. Call a graph equitably $k$-choosable if it has an equitable $L$-coloring for every $k$-list assignment $L$. A graph $G$ is $(a,b)$-sparse if for every $A\subseteq V(G)$, the number of edges in the subgraph $G[A]$ of $G$ induced by $A$ is at most $a|A|+b$. Our first main result is that every $(\frac{7}{6},\frac{1}{3})$-sparse graph with minimum degree at least $2$ is equitably $3$-colorable and equitably $3$-choosable. This is sharp. Our second main result is that every $(\frac{5}{4},\frac{1}{2})$-sparse graph with minimum degree at least $2$ is equitably $4$-colorable and equitably $4$-choosable. This is also sharp. One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.

Figures (4)

• Figure 3.1: $\rho_{G}^{3}=3$ and $G$ cannot be equitably $3$ colored, where $G=G_{1,0}$ on the left, $G=G_{1,2}$ in the middle and $G=G_{1,5}$ on the right.
• Figure 3.2: $\rho_{G}^{4}=3-\sigma_{G}^{4}$ and $G$ cannot be equitably $4$ colored, where $G=G_{1,0}$ on the left, $G=G_{1,3}$ in the middle and $G=G_{1,7}$ on the right.
• Figure 4.1: The figure shows four maximal bugs. In (a), $v_1,v_2$ are leg vertices, $v_3,v_4$ are body vertices, and $v_3,v_4,r$ are hidden; in (b), $v_4$ is a body vertex and $v_4,r$ are hidden; in (c), $v_1,v_2,x,y$ are leg vertices, and $x,r$ are hidden; in (d), $v_1,v_2,x,y$ are leg vertices, and $x$, but not $r$, is hidden.
• Figure 6.1: The maximal bug $B(r)$ is above the dotted line; the buffer $F$ is a forest with two components; the core $H$ has three components, one of which contains the path $P$ ending in the only leaf $\ell$ of $G$.

Theorems & Definitions (44)

• Theorem 1.1: Luo, Sereni, Stephens and Yu
• Theorem 1.2: Dong and Zhang
• Theorem 1.3
• Theorem 2.1
• Example 3.1
• proof
• Example 3.2
• proof
• Lemma 4.1
• proof