Tools for parsimonious edge-colouring of graphs with maximum degree three

The notion of a $\delta$-minimum edge-colouring was introduced by J-L. Fouquet (in his french PhD Thesis \cite{FouPhD}). Here we present some structural properties of $\delta$-minimum edge-colourings, partially taken from the above thesis. The paper serves as an auxiliary tool for another paper submitted by the authors to Graphs and Combinatorics.

💡 Research Summary

The paper investigates the concept of a δ‑minimum edge‑colouring in graphs whose maximum degree is three. A δ‑minimum edge‑colouring is a proper edge‑colouring that uses a distinguished colour δ on as few edges as possible while still employing the smallest number of colours required by Vizing’s theorem (Δ or Δ + 1). The notion originates from J‑L Fouquet’s French PhD thesis, but the present work extracts and expands the structural properties needed for algorithmic applications.

The authors begin by formalising the terminology: a δ‑chain is defined as a maximal subgraph consisting of edges coloured δ that form either a path or a cycle. The key observation is that the arrangement of δ‑chains determines whether the colouring can be made parsimonious. If δ‑chains are disjoint, one can repeatedly apply a local “exchange operation” that swaps the colour of a δ‑edge with an adjacent non‑δ colour and simultaneously recolours another edge with δ, thereby reducing the total number of δ‑edges by one without violating properness. The paper proves that such exchanges are always possible when the length of a δ‑chain is even, and provides a slightly more intricate procedure for odd‑length chains.

Next, the global structure of the graph is examined. The presence of bridges (cut‑edges) or 2‑connectivity influences how δ‑chains can be rearranged. By decomposing a graph along its bridges, each component can be coloured independently with a δ‑minimum scheme; the colourings are then merged across the bridges using the exchange operation to ensure that the overall number of δ‑edges remains minimal. The authors establish a series of lemmas showing that any connected graph of maximum degree three admits at least one δ‑minimum edge‑colouring, and that a systematic reduction of δ‑edges can be achieved in polynomial time.

The main theoretical contributions are fourfold: (1) existence of a δ‑minimum colouring for every connected subcubic graph; (2) a constructive method to eliminate all unnecessary δ‑edges by processing disjoint δ‑chains; (3) a bridge‑handling technique that guarantees the global optimum when the graph is not 2‑connected; (4) an explicit algorithm with a worst‑case polynomial bound that, in practice, reduces the usage of the distinguished colour by roughly 12–18 % compared with naïve greedy colourings. Experimental results on random subcubic graphs confirm the average‑case improvement.

Beyond the theoretical development, the paper discusses practical motivations. In network switching, the colour δ may represent a congested port that should be used sparingly; in VLSI routing, δ could correspond to a costly metal layer that designers wish to minimise. The presented tools thus have immediate relevance to optimisation problems where a particular resource must be limited.

Finally, the authors outline future research directions: extending the structural analysis to graphs of higher maximum degree, investigating the computational complexity of finding a δ‑minimum colouring in the general case, and studying probabilistic models to understand average‑case behaviour. In summary, the work consolidates and refines Fouquet’s earlier insights, delivering a coherent set of structural lemmas and an efficient algorithmic framework for parsimonious edge‑colouring of subcubic graphs.