List Edge‐Coloring and Total Coloring in Graphs of Low Treewidth

Abstract

Artículo de publicación ISIWe prove that the list chromatic index of a graph of maximum degree Delta and treewidth <= root 2 Delta -3 is Delta; and that the total chromatic number of a graph of maximum degree and treewidth <= Delta/3+1 is Delta+1. This improves results by Meeks and Scott.Fondation Sciences Mathematiques de Paris 11090141 Fondecyt 114076

The linear arboricity of graphs with low treewidth

“…By the lemma, we have δ(G) ≥ 2. At the same time, we may apply Lemma 3 in [5] with parameters ∆ 0 = 2t, and obtain the following result.…”

“…Lemma 4 [5]. There are disjoint vertex sets U, W ⊆ V (G) and a vertex x ∈ U , such that (a) W is stable with…”

“…In [5], it is proved that if G is a graph with ∆ ≥ 3k − 3 and k ≥ 3, then the total chromatic χ ′′ (G) = ∆ + 1. In this paper, we show that if ∆ ≥ 3k − 3 and k = 3 or k = 4, then the linear arboricity la(G) is ∆ 2 .…”

Chromatic index, treewidth and maximum degree

“…Isobe et al [10] show that any k-degenerate graph of maximum degree ∆ ≥ 4k + 3 has a total colouring with only ∆ + 1 colours. For graphs that are not only k-degenerate but also of treewidth k, a maximum degree of ∆ ≥ 3k − 3 already suffices [4]. Noting that they are 5-degenerate, we include some results on planar graphs as well.…”