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Graphs with given automorphism group and large clique number

John Haslegrave

Abstract

Barbieri recently showed that the finite graphs realising any given finite automorphism group have unbounded genus, answering a question of Cornwell et al. In this note we give a short proof of a stronger result: they have unbounded clique number.

Graphs with given automorphism group and large clique number

Abstract

Barbieri recently showed that the finite graphs realising any given finite automorphism group have unbounded genus, answering a question of Cornwell et al. In this note we give a short proof of a stronger result: they have unbounded clique number.
Paper Structure (2 theorems)

This paper contains 2 theorems.

Key Result

Proposition 1

Let $G$ be a non-complete finite graph, and let $G '$ be the graph obtained from $G$ by adding a disjoint clique with the same number of vertices and a perfect matching between the two. Then $\operatorname{Aut}( G ')\cong \operatorname{Aut}( G )$.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Corollary 2
  • proof