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Strictly critical snarks with girth or cyclic connectivity equal to 6

Ján Mazák, Jozef Rajník, Martin Škoviera

Abstract

A snark -- connected cubic graph with chromatic index $4$ -- is critical if the graph resulting from the removal of any pair of distinct adjacent vertices is $3$-edge-colourable; it is bicritical if the same is true for any pair of distinct vertices. A snark is strictly critical if it is critical but not bicritical. Very little is known about strictly critical snarks. Computational evidence suggests that strictly critical snarks constitute a tiny minority of all critical snarks. Strictly critical snarks of order $n$ exist if and only if $n$ is even and at least 32, and for each such order there is at least one strictly critical snark with cyclic connectivity $4$. A sparse infinite family of cyclically $5$-connected strictly critical snarks is also known, but those with cyclic connectivity greater than $5$ have not been discovered so far. In this paper we fill the gap by constructing cyclically $6$-connected strictly critical snarks of each even order $n\ge 342$. In addition, we construct cyclically $5$-connected strictly critical snarks of girth 6 for every even $n\ge 66$ with $n\equiv 2\pmod8$.

Strictly critical snarks with girth or cyclic connectivity equal to 6

Abstract

A snark -- connected cubic graph with chromatic index -- is critical if the graph resulting from the removal of any pair of distinct adjacent vertices is -edge-colourable; it is bicritical if the same is true for any pair of distinct vertices. A snark is strictly critical if it is critical but not bicritical. Very little is known about strictly critical snarks. Computational evidence suggests that strictly critical snarks constitute a tiny minority of all critical snarks. Strictly critical snarks of order exist if and only if is even and at least 32, and for each such order there is at least one strictly critical snark with cyclic connectivity . A sparse infinite family of cyclically -connected strictly critical snarks is also known, but those with cyclic connectivity greater than have not been discovered so far. In this paper we fill the gap by constructing cyclically -connected strictly critical snarks of each even order . In addition, we construct cyclically -connected strictly critical snarks of girth 6 for every even with .

Paper Structure

This paper contains 13 sections, 16 theorems, 10 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cyclically 5-connected strictly critical snarks of girth 6
  4. Case (i).
  5. Case (ii).
  6. Case (iii).
  7. Case (iv).
  8. Cyclically 6-connected strictly critical snarks
  9. Case (i).
  10. Case (ii).
  11. Case (iii).
  12. Case (iv).
  13. Case (v).

Key Result

Theorem 1.1

There exists a strictly critical snark of order $n$ for each even $n\ge 32$.

Figures (16)

  • Figure 1: A strictly critical snark of the smallest possible order $32$ with the only removable pair $\{x, y\}$ of vertices.
  • Figure 2: A $(2,2,2)$-pole found in cyclically $5$-connected strictly critical snarks of order 36
  • Figure 3: The structure of the snark $H_6*\operatorname{TTT}_{\text{sc}}(T_1, T_2, T_3)$
  • Figure 4: Colouring for Case (i)
  • Figure 5: Colouring for Case (ii)
  • ...and 11 more figures

Theorems & Definitions (32)

  • Theorem 1.1: Chladny-Skoviera-Factorisations
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Parity Lemma Descartes
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3: Chladný and Škoviera Chladny-Skoviera-Factorisations
  • Definition 3.4
  • Lemma 3.5
  • ...and 22 more