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Regular colouring defect of a cubic graph and the conjectures of Fan-Raspaud and Fulkerson

Ján Karabáš, Edita Máčajová, Roman Nedela, Martin Škoviera

TL;DR

This work introduces the regular colouring defect $rdf(G)$ for bridgeless cubic graphs and studies its relationship to the colouring defect $df(G)$ and to the Fan–Raspaud and Fulkerson conjectures. The authors relate $rdf(G)$ to nowhere-zero flows and Fano-plane configurations, and show that $rdf(G)≥df(G)$ with equivalences linking regular arrays to Fulkerson covers. They prove the main result that the gap between $df(G)$ and $rdf(G)$ can be arbitrarily large: for each $d≥4$ there exists a graph with $df(G)=4$ and $rdf(G)≥d$, using Petersen-based superposition and inflation of adjacent vertices to triangles to control the invariants. The findings illuminate structural connections between edge-colourings, flows, and covers, and raise open questions about snarks with high regular defect and connectivity.

Abstract

We introduce a new invariant of a cubic graph - its regular colouring defect - which is defined as the smallest number of edges left uncovered by any collection of three perfect matchings that have no edge in common. This invariant is a modification of colouring defect, an invariant introduced by Steffen (J. Graph Theory 78(2015), 195--206), whose definition does not require the empty intersection condition. In this paper we discuss the relationship of this invariant to the well-known conjectures of Fulkerson (1971) and Fan and Raspaud (1994) and prove that colouring defect and regular colouring defect can be arbitrarily far apart.

Regular colouring defect of a cubic graph and the conjectures of Fan-Raspaud and Fulkerson

TL;DR

This work introduces the regular colouring defect for bridgeless cubic graphs and studies its relationship to the colouring defect and to the Fan–Raspaud and Fulkerson conjectures. The authors relate to nowhere-zero flows and Fano-plane configurations, and show that with equivalences linking regular arrays to Fulkerson covers. They prove the main result that the gap between and can be arbitrarily large: for each there exists a graph with and , using Petersen-based superposition and inflation of adjacent vertices to triangles to control the invariants. The findings illuminate structural connections between edge-colourings, flows, and covers, and raise open questions about snarks with high regular defect and connectivity.

Abstract

We introduce a new invariant of a cubic graph - its regular colouring defect - which is defined as the smallest number of edges left uncovered by any collection of three perfect matchings that have no edge in common. This invariant is a modification of colouring defect, an invariant introduced by Steffen (J. Graph Theory 78(2015), 195--206), whose definition does not require the empty intersection condition. In this paper we discuss the relationship of this invariant to the well-known conjectures of Fulkerson (1971) and Fan and Raspaud (1994) and prove that colouring defect and regular colouring defect can be arbitrarily far apart.
Paper Structure (5 sections, 8 theorems, 4 equations, 3 figures)

This paper contains 5 sections, 8 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular colouring defect of a cubic graph
  4. Proof of Theorem \ref{['thm:main']}
  5. Remarks

Key Result

Theorem 1.1

For every integer $d\ge 4$ there exists a bridgeless cubic graph $G$ whose colouring defect equals $4$ and regular colouring defect is at least $d$.

Figures (3)

  • Figure 1: Four lines of the Fano plane that correspond to regular $3$-arrays
  • Figure 2: An irregular $4$-core with edge-labels showing the number of perfect matchings containing the edge in question
  • Figure 3: The superedge $F_g$ (left) and the resulting snark $G_g$ (right)

Theorems & Definitions (14)

  • Theorem 1.1
  • Lemma 2.1: Parity Lemma
  • Proposition 3.1: KMNS-red
  • Corollary 3.2
  • proof
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Proposition 4.1
  • proof
  • ...and 4 more