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arXiv · 1809.08281

Equitable List Vertex Colourability and Arboricity of Grids

Ewa Drgas-Burchardt·Janusz Dybizbański·Hanna Furmańczyk·Elzbieta Sidorowicz
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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 $d\geq 5$.

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Ewa Drgas-Burchardt, Janusz Dybizbański, Hanna Furmańczyk, Elzbieta Sidorowicz. 2018-09-21. Equitable List Vertex Colourability and Arboricity of Grids. https://doi.org/10.2298/fil1818353d

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