Equitable list vertex colourability and arboricity of grids
Ewa Drgas‐Burchardt, Janusz Dybizbański, Hanna Furmańczyk, Elżbieta Sidorowicz · Filomat · 2018
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 ?=l (G). Zhang in 2016 posed the conjecture that if k ? ?(?(G) + 1)/2? 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 2 {2,3,4} and give new bounds on ?=l (G) for general graphs and for d-dimensional grids with d ? 5.