Equitable list point arboricity of graphs

Abstract

A graph G is list point k-arborable if, whenever we are given a k-list assignment L(v) of colors for each vertex v ∈ V(G), we can choose a color c(v) ∈ L(v) for each vertex v so that each color class induces an acyclic subgraph of G, and is equitable list point k-arborable if G is list point k-arborable and each color appears on at most ⌈|V(G)|/k⌉ vertices of G. In this paper, we conjecture that every graph G is equitable list point k-arborable for every k ≥ ⌈(∆(G)+1)/2⌉ and settle this for complete graphs, 2-degenerate graphs, 3-degenerate claw-free graphs with maximum degree at least 4, and planar graphs with maximum degree at least 8.

Equitable list vertex colourability and arboricity of grids

“…A graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. Since every 2-degenerate graph has arboricity 2, Theorem 2.11 confirms the result for 2-degenerate graphs obtained by Zhang [13].…”

“…Zhang [13] confirmed above two conjectures for complete graphs, 2-degenerate graphs, 3-degenerate claw-free graphs with maximum degree at least 4, and planar graphs with maximum degree at least 8. Our results confirm above conjectures for some Cartesian products of paths, i.e.…”

Equitable d-degenerate Choosability of Graphs

“…Sometimes we have some additional requirements on vertices/nodes that can be modeled by a list of available colours. So, we are interested in the list version, introduced by Kostochka, Pelsmajer and West [5] (an independent case), and by Zhang [10] (an acyclic case).…”

“…Given k ∈ N and d ∈ N 0 , a graph G is equitably (k, D d )-choosable if for any k-uniform list assignment L there is an (L, D d )-colouring of G such that the size of any colour class does not exceed |V (G)|/k . The notion of equitable (k, D 0 )-choosability was introduced by Kostochka et al [5] whereas the notation of equitable (k, D 1 )-choosability was introduced by Zhang [10].…”

“…For S ⊆ V (G) by G−S we denote a subgraph of G induced by V (G)\S. We start with a generalization of some results given in [5,9,10] for classes D 0 and D 1 .…”