Equitable List Vertex Colourability and Arboricity of Grids

TL;DR

This paper investigates equitable list vertex arboricity in graphs, proves Zhang's conjecture for 2-4 dimensional grids, and provides new bounds for higher-dimensional grids and general graphs.

Contribution

It introduces new tools for analyzing equitable list arborability and confirms Zhang's conjecture for certain grid classes, advancing understanding of graph coloring properties.

Findings

Proves Zhang's conjecture for 2-, 3-, and 4-dimensional grids.

Provides new bounds on equitable list vertex arboricity for higher-dimensional grids.

Develops methods applicable to general graphs for determining arborability.

Abstract

A graph $G$ is equitably $k$-list arborable if for any $k$-uniform list assignment $L$, there is an equitable $L$-colouring of $G$ whose each colour class induces an acyclic graph. The smallest number $k$ admitting such a coloring is named equitable list vertex arboricity and is denoted by $\rho_l^=(G)$. Zhang in 2016 posed the conjecture that if $k \geq \lceil (\Delta(G)+1)/2 \rceil$ then $G$ is equitably $k$-list arborable. We give some new tools that are helpful in determining values of $k$ for which a general graph is equitably $k$-list arborable. We use them to prove the Zhang's conjecture for $d$-dimensional grids where $d \in \{2,3,4\}$ and give new bounds on $\rho_l^=(G)$ for general graphs and for $d$-dimensional grids with $d\geq 5$.