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Extremal regular graphs of given chromatic number

Christian Rubio-Montiel

Abstract

We define an extremal $(r|χ)$-graph as an $r$-regular graph with chromatic number $χ$ of minimum order. We show that the Tur{\' a}n graphs $T_{ak,k}$, the antihole graphs and the graphs $K_k\times K_2$ are extremal in this sense. We also study extremal Cayley $(r|χ)$-graphs and we exhibit several $(r|χ)$-graph constructions arising from Tur{\' a}n graphs.

Extremal regular graphs of given chromatic number

Abstract

We define an extremal -graph as an -regular graph with chromatic number of minimum order. We show that the Tur{\' a}n graphs , the antihole graphs and the graphs are extremal in this sense. We also study extremal Cayley -graphs and we exhibit several -graph constructions arising from Tur{\' a}n graphs.

Paper Structure

This paper contains 12 sections, 15 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. Cayley $(r|\chi)$-graphs
  3. Antihole graphs
  4. The case of $r=\chi$
  5. Non-Cayley constructions
  6. Upper bound
  7. The graph $T^*_{n,\chi}$
  8. The graph $G_{a,c,t}$
  9. Small values
  10. Extremal $(5|3)$-graph
  11. Extremal $(7|\chi)$-graphs for $\chi=3,6$
  12. Extremal $(9|3)$-graph

Key Result

Lemma 2.1

The $(ak,k)$-Turán graph $T_{ak,k}$ is a Cayley graph.

Theorems & Definitions (19)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • ...and 9 more