[Paper Review] Tools for parsimonious edge-colouring of graphs with maximum degree three

Jean‐Luc Fouquet, Jean-Marie Vanherpe|arXiv (Cornell University)|Jan 28, 2012

Advanced Graph Theory Research4 references3 citations

TL;DR

This paper presents structural properties of δ-minimum edge-colourings in graphs with maximum degree three, where δ is used as sparingly as possible. It establishes that such colourings are proper, characterizes the configuration of δ-coloured edges via alternating paths in other colour pairs, and proves that δ-edges are constrained by vertex degrees and path connectivity, leading to tight bounds on their arrangement and interactions in the graph.

ABSTRACT

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.

Motivation & Objective

• To characterize δ-minimum edge-colourings in graphs of maximum degree three, where δ is used as few times as possible.
• To establish that δ-minimal colourings are necessarily proper, resolving ambiguity in improper colouring approaches.
• To analyze the structural constraints on edges coloured δ, particularly their adjacency to all other colours and degree conditions on endpoints.
• To determine how δ-coloured edges relate to alternating paths in other colour pairs (α,β), (β,γ), (α,γ), and how these paths influence minimality.
• To derive bounds on the number of edges that can be mutually connected via δ-coloured edges, especially under shared path connectivity constraints.

Proposed method

• Define δ-minimum edge-colouring as a proper edge-colouring using four colours (α,β,γ,δ) where the number of δ-coloured edges, s(G), is minimized.
• Introduce the sets Aϕ, Bϕ, Cϕ to classify δ-coloured edges based on their connection via alternating paths in (α,β), (β,γ), (α,γ) subgraphs.
• Use path exchange techniques (colour swapping along alternating paths) to prove minimality and structural invariance under reconfiguration.
• Apply contradiction arguments by recolouring δ-edges to other colours when possible, showing that δ-minimality forces specific structural constraints.
• Leverage the structure of 2-factors and odd cycles to prove lower bounds on the number of odd cycles in any 2-factor, linking to s(G).
• Use vertex degree constraints (2 or 3) on endpoints of δ-edges to restrict possible configurations and enforce path alternation.

Experimental results

Research questions

• RQ1 What structural constraints must δ-coloured edges satisfy in a δ-minimum edge-colouring of a maximum-degree-3 graph?
• RQ2 Can a δ-minimum edge-colouring be improper, or must it always be proper?
• RQ3 How do alternating paths in (α,β), (β,γ), and (α,γ) subgraphs constrain the placement and connectivity of δ-coloured edges?
• RQ4 What is the maximum number of edges that can be mutually connected via δ-coloured edges under shared path membership in Aϕ, Bϕ, or Cϕ?
• RQ5 Under what conditions can two δ-coloured edges be adjacent or connected by a path, and what configurations are forbidden?

Key findings

• Any δ-minimum edge-colouring is necessarily proper, as any improper use of δ could be reduced by recolouring.
• Each δ-coloured edge is incident to all three colours α, β, γ, and has one or both endpoints of degree 3.
• For any δ-coloured edge e, there exists a path of even length in one of the (α,β), (β,γ), or (α,γ) subgraphs connecting its endpoints, with the path lying entirely in the corresponding Cϕ(e) set.
• If two δ-coloured edges e₁ and e₂ belong to different sets (e.g., Aϕ and Bϕ), then they are vertex-disjoint and induce a 2K₂ subgraph.
• If three δ-coloured edges belong to the same set (e.g., Aϕ), then they induce a subgraph with at most four edges, and any attempt to add more edges leads to a contradiction with δ-minimality.
• The cycles Cϕ(e) associated with each δ-coloured edge e are vertex-disjoint across different e, and their colour sequences can be permuted to reassign δ to any other edge while preserving minimality.