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Circular chromatic index of small graphs

Ján Mazák, Filip Zrubák

Abstract

We systematically determine circular chromatic index of small graphs and multigraphs with maximum degree $4$, $5$, $6$ (and also their number for a given small order). We construct several infinite families of such graphs with circular chromatic index in the set $\{Δ+ 1/2, Δ+ 2/3, Δ+ 3/4$, $Δ+ 1\}$. Our results refute edge-connectivity variants of the ``Upper Gap Conjecture'' (about the non-existence of graphs with circular chromatic index just below $Δ+ 1$).

Circular chromatic index of small graphs

Abstract

We systematically determine circular chromatic index of small graphs and multigraphs with maximum degree , , (and also their number for a given small order). We construct several infinite families of such graphs with circular chromatic index in the set , . Our results refute edge-connectivity variants of the ``Upper Gap Conjecture'' (about the non-existence of graphs with circular chromatic index just below ).
Paper Structure (8 sections, 13 theorems, 3 equations, 6 figures, 9 tables)

This paper contains 8 sections, 13 theorems, 3 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Computational methods
  3. Overview of values of $\chi_c'$
  4. Graphs with large values of $\chi_c'$
  5. Families with large values of $\chi_c'$ for $\Delta = 6$
  6. Families with large values of $\chi_c'$ for $\Delta = 5$
  7. Families with large values of $\chi_c'$ for $\Delta = 4$
  8. Open problems

Key Result

Lemma 1

Assume that there exists a connected graph $H$ with maximum degree $\Delta$, $\chi_c'(H) = \Delta + 1$, exactly two vertices of degree $1$, and having no bridge except those incident with vertices of degree $1$. Then there exist infinitely many $\Delta$-regular $2$-edge-connected graphs $G$ satisfyi

Figures (6)

  • Figure 1: Subgraph $H$ of $C_{2n+1}(1,2)$ drawn with equilateral triangles.
  • Figure 2: All $12$ possible colourings of $H$.
  • Figure 3: Contradiction for $\varepsilon < 1/2$ in $C_{2n+1}(1,2)$.
  • Figure 4: $(9,2)$-edge-colouring of $G_{13}$
  • Figure 5: $(9,2)$-edge-colouring of $G_{15}$
  • ...and 1 more figures

Theorems & Definitions (26)

  • Conjecture 1
  • Conjecture 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 16 more