Cubic graphs of colouring defect 3 and conjectures of Berge and Alon-Tarsi

TL;DR

The work links two central uncolourability measures for bridgeless cubic graphs, colouring defect $\mathrm{df}(G)$ and perfect matching index $\pi(G)$, to long-standing conjectures in cycle covers. It provides a complete structural characterization of $\mathrm{df}(G)=3$ graphs with $\pi(G)\ge5$, showing they originate from the Petersen graph via controlled insertions and vertex substitutions on a fixed $6$-cycle, and uses this to derive tight cycle-cover bounds $\ell(\mathcal{C})\le\tfrac{4}{3}m+1$ for such graphs. The paper also develops decomposition tools and a theory of hexagonal cores and essential triangles to analyze snarks with defect $3$, proving that many such graphs admit a 4-edge-cover and enumerating cases where $\pi(G)=5$. These results reinforce the connections between perfect-matching covers, cycle covers, and 3-edge-colourability, and provide a concrete, Petersen-based framework for understanding the Berge and Alon–Tarsi conjectures in a broad class of cubic graphs.

Abstract

We study two measures of uncolourability of cubic graphs, their colouring defect and perfect matching index. The colouring defect of a cubic graph $G$ is the smallest number of edges left uncovered by three perfect matchings; the perfect matching index of $G$ is the smallest number of perfect matchings that together cover all edges of $G$. We provide a complete characterisation of cubic graphs with colouring defect $3$ whose perfect matching index is greater or equal to $5$. The result states that every such graph arises from the Petersen graph with a fixed $6$-cycle $C$ by substituting edges or vertices outside $C$ with suitable $3$-edge-colourable cubic graphs. Our research is motivated by two deep and long-standing conjectures, Berge's conjecture stating that five perfect matchings are enough to cover the edges of any bridgeless cubic graph and the shortest cycle cover conjecture of Alon and Tarsi suggesting that every bridgeless graph can have its edges covered with cycles of total length at most $7/5\cdot m$, where $m$ is the number of edges. We apply our characterisation to showing that every cubic graph with colouring defect $3$ admits a cycle cover of length at most $4/3\cdot m +1$, where $m$ is the number of edges, the bound being achieved by the graphs whose perfect matching index equals $5$. We further prove that every snark containing a $5$-cycle with an edge whose endvertices removed yield a $3$-edge-colourable graph has a cycle cover of length at most $4/3\cdot m+1$, as well.

Figures (14)

• Figure 1: Four lines of the Fano plane forming the configuration $F_4$.
• Figure 2: An optimal $3$-array of the Petersen graph; its core is displayed bold.
• Figure 3: The configuration $F_4$ for $3$-arrays with no triply covered edge.
• Figure 4: A hexagonal core in a snark and its immediate neighbourhood.
• Figure 5: A triangle intersected by a hexagonal core.

Theorems & Definitions (47)

• Lemma 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• proof
• Corollary 3.6
• Theorem 4.1
• proof