Edge-Coloring Partialk-Trees

Chromatic index, treewidth and maximum degree

“…Vizing [16] (see also Zhou et al [18]) proved that every graph of treewidth k and maximum degree ∆ ≥ 2k has an edge-colouring with ∆ colours. Proposition 1 shows that this bound is not tight.…”

“…Vizing [16] (see also Zhou et al [18]) observed a consequence of his adjacency lemma: any graph with treewidth k and maximum degree at least 2k has chromatic index χ ′ (G) = ∆(G). 1 Is this tight?…”

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

“…In the introduction we mentioned the result of Zhou et al. that whenever is at least twice the treewidth. If one believes the list edge‐coloring conjecture then this indicates that in Theorem a maximum degree that is linear in k is already sufficient to guarantee the assertion.…”

“…Our proofs rely on the fact that graphs with low treewidth and a high maximum degree contain substructures that are suitable for classical coloring arguments. This method has been used before: Zhou et al [17] show that χ (G) = (G) if the graph G has treewidth ≤ 1 2 (G); Juvan et al [9] prove that the edges of any graph of treewidth 2 can be colored from lists of size ; and in [11] the latter results are extended to graphs of treewidth 3 and maximum degree ≥ 7. Finally, this approach has also been employed by Meeks and Scott [12], who prove that determining the list chromatic index as well as the list total chromatic number is fixed parameter tractable, when parameterized by treewidth.…”

“…This method has been used before: Zhou et al. show that if the graph G has treewidth ; Juvan et al. prove that the edges of any graph of treewidth 2 can be colored from lists of size Δ; and in the latter results are extended to graphs of treewidth 3 and maximum degree ⩾7.…”