On parsimonious edge-colouring of graphs with maximum degree three

In a graph $G$ of maximum degree $\Delta$ let $\gamma$ denote the largest fraction of edges that can be $\Delta$ edge-coloured. Albertson and Haas showed that $\gamma \geq 13/15$ when $G$ is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for which $\gamma = 13/15$. This extends a result given by Steffen. These results are obtained by using structural properties of the so called $\delta$-minimum edge colourings for graphs with maximum degree 3. Keywords : Cubic graph; Edge-colouring

💡 Research Summary

The paper studies edge‑colourings of connected graphs whose maximum degree Δ equals three, focusing on colourings that minimise the use of a distinguished colour δ. By Vizing’s theorem such graphs are 4‑edge‑colourable, and a proper edge‑colouring using colours {α,β,γ,δ} can always be chosen. The authors define a δ‑minimum edge‑colouring as one that uses the colour δ on as few edges as possible; the number of δ‑edges is denoted s(G).

Section 2 establishes basic lemmas about δ‑minimum colourings. Lemma 1 shows that any δ‑improper colouring can be transformed into a proper one without increasing the set of δ‑edges, guaranteeing the existence of a δ‑minimum colouring. Proposition 2 and Lemma 3 relate s(G) to the size of a vertex set whose removal makes the graph 3‑edge‑colourable, proving that s(G) is at most the size of such a set.

The central structural result is Theorem 4. It partitions the δ‑edges into three families Aφ, Bφ, Cφ. Each edge e∈Aφ (resp. Bφ, Cφ) lies on a unique odd cycle CA(e) (resp. CB(e), CC(e)) that contains exactly one δ‑edge (namely e) and whose remaining edges alternate between two of the colours {α,β,γ}. These cycles are vertex‑disjoint, any two cycles belonging to distinct families are at distance at least two, and a family can contain at most three δ‑edges, which together induce at most four edges. Moreover, no two consecutive vertices of a cycle have degree two, and vertices of degree two are isolated in a precise way. These constraints heavily restrict how many δ‑edges a graph can have.

Lemma 5 refines the placement of a δ‑edge according to the degrees of its endpoints, showing that when one endpoint has degree two the edge may belong to two families simultaneously. Lemma 6, specialized to cubic graphs, proves that any two cycles from the same family are either at distance ≥2, joined by at least three edges, or share at least two vertices. Lemma 7 demonstrates that any two distinct δ‑edges induce a 2K₂ subgraph, i.e., they are far apart in the graph.

Section 3 applies the structural theory. In 3.1 the authors revisit a result of Payan concerning strong matchings. Theorem 8 proves that a δ‑minimum colouring can be chosen so that the set of δ‑edges forms a strong matching (no two edges share a vertex or are adjacent) and each such edge has both ends of degree three. Consequently, Corollary 9 shows that removing s(G) vertices (one endpoint of each δ‑edge) leaves a 3‑edge‑colourable graph. This recovers and extends Steffen’s earlier work on cubic graphs.

Section 3.2 addresses parsimonious edge‑colouring, i.e., maximizing the fraction γ(G) of edges that can be coloured with only three colours. The authors define c(G) as the size of the largest subgraph with chromatic index three and set γ(G)=c(G)/|E(G)|. Lemma 10 establishes the simple identity γ(G)=1−s(G)/m, where m=|E(G)|. Using the structural properties of δ‑minimum colourings, Theorem 11 derives a general lower bound γ(G)≥1−2/(3·g₀(G)), where g₀(G) is the odd girth (length of the shortest odd cycle). This bound improves earlier results of Albertson‑Haas (γ≥13/15 for cubic graphs) and Bondy‑Locke (γ≥26/31 for Δ≤3).

The authors then identify the extremal graphs attaining the bound γ=13/15. Apart from the Petersen graph, there is exactly one other graph (the 5‑vertex graph G₅ shown in Figure 1) that reaches this value; all other Δ≤3 graphs satisfy γ>13/15. Lemma 12 shows that deleting a degree‑one vertex strictly increases γ, while Lemma 13 proves that shrinking a reducible triangle also increases γ. Finally, Theorem 14 gives a refined bound involving the numbers of degree‑2 and degree‑3 vertices, namely γ(G)≥1−2/(15·(1+2/3·|V₂|·|V₃|)), again with equality only for the two extremal graphs.

In conclusion, the paper extends Albertson and Haas’s 13/15 bound from cubic graphs to all graphs of maximum degree three (except the small exceptional graph G₅), identifies precisely the two graphs where the bound is tight, and does so by a detailed analysis of δ‑minimum edge‑colourings. The work unifies and strengthens earlier results by Steffen, Payan, and others, and provides a clear structural framework that may be useful for further investigations into edge‑colouring problems on low‑degree graphs.