Abstract
We conjecture that any graph $G$ with treewidth $k$ and maximum degree $\Delta(G)\geq k + \sqrt{k}$ satisfies $\chi'(G)=\Delta(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth $k\geq 4$ and maximum degree $2k-1$ satisfies $\chi'(G)=\Delta(G)$, extending an old result of Vizing.
Highlights
The least number χ′(G) of colours necessary to properly colour the edges of a graph G is either the maximum degree ∆(G) or ∆(G)+1
Becaus√e we can exclude that graphs with treewidth k and maximum degree ∆ ≥ k + k are overfull, the overfull conjecture implies that such graphs on less than 3∆ vertices can always be edge-coloured with ∆ colours
If (T, B) is a tree decomposition of the graph G, for any vertex v of G we denote by T (v) the subtree of T that consists of those vertices corresponding to bags that contain v
Summary
The least number χ′(G) of colours necessary to properly colour the edges of a (simple) graph G is either the maximum degree ∆(G) or ∆(G)+1. Vizing [16] (see 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).. The bound is close to best possible: in Section 5 we construc√t, for infini√tely many k, graphs with treewidth k, maximum degree ∆ = k + ⌊ k⌋ < k + k, and chromatic index ∆ + 1. Becaus√e we can exclude that graphs with treewidth k and maximum degree ∆ ≥ k + k are overfull, the overfull conjecture (as well as our conjecture) implies that such graphs on less than 3∆ vertices can always be edge-coloured with ∆ colours. Vizing [16] originally showed that k-degenerate graphs, rather than treewidth k graphs, of maximum degree ∆ ≥ 2k have an edge-colouring with ∆ colours. Sanders and Zhao [13] and independently Zhang [17] extended this to ∆(G) ≥ 7
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