On a limit of the method of Tashkinov trees for edge-colouring

Abstract

The main technique used to edge-colour graphs requiring ∆ + 2 or more colours is the method of Tashkinov trees. We present a specific limit to this method, in terms of Kempe changes. We also provide a new Tashkinov tree extension.

Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

“…. Therefore w i cannot be the supporting vertex of T it with respect to ϕ it and connecting color δ it (see Algorithm 3.1); this contradiction proves (2).…”

“…As remarked by McDonald [23], the Goldberg-Seymour conjecture and ideas culminating in this method are two cornerstones in modern edge-coloring. Nevertheless, this method suffers some theoretical limitation when applied to prove the conjecture; the reader is referred to Asplund and McDonald [2] for detailed information. Despite various attempts to extend the Tashkinov trees (see, for instance, [3,4,5,31,36]), the difficulty encountered by the method remains unresolved.…”

Densities, Matchings, and Fractional Edge-Colorings

“…. , T p be components of G\S, with p > |S|, |T i | odd, and (G); this contradiction establishes (1). Thus Γ w (G) > ∆ w (G), contradicting the hypothesis of this theorem.…”

“…If |X| ≥ 3, then U k+1 = X is an optimal solution to (3.4) and hence to (3.3). (1) and 3, we see that (4) z(U, α k ) ≤ 0 for any U ⊆ V with |U | odd. Since z(U k , α k ) = 2w(U k ) − α k (|U k | − 1) = 0 by the definition of α k and |U k | ≥ 3, from (4) we deduce that U k+1 = U k is an optimal solution to (3.3).…”

“…The crux of this technique is to capture the density Γ(G) required to prove Conjecture 1.1, by exploring a sufficiently large tree, the so-called Tashkinov tree. However, this target may become unreachable when χ ′ (G) gets close to ∆(G), even if we allow for an unlimited number of Kempe changes; such an example can be found in Asplund and McDonald [1]. Therefore it is desirable to have some new approaches to multigraph edge-coloring.…”