Tetravalent edge-transitive graphs of girth at most 4

Primož Potočnik, Steve Wilson

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

Abstract

This is the first in the series of articles stemming from a systematical investigation of finite edge-transitive tetravalent graphs, undertaken recently by the authors. In this article, we study a special but important case in which the girth of such graphs is at most 4. In particular, we show that, except for a single arc-transitive graph on 14 vertices, every edge-transitive tetravalent graph of girth at most 4 is the skeleton of an arc-transitive map on the torus or has one of these two properties: (1)there exist two vertices sharing the same neighbourhood,(2)every edge belongs to exactly one girth cycle. Graphs with property (1) or (2) are then studied further. It is shown that they all arise either as subdivided doubles of smaller arc-transitive tetravalent graphs, or as line graphs of triangle-free (G, 1)-regular and (G, 2)-arc-transitive cubic graphs, or as partial line graphs of certain cycle decompositions of smaller tetravalent graphs.

Original languageEnglish (US)
Pages (from-to)217-236
Number of pages20
JournalJournal of Combinatorial Theory. Series B
Volume97
Issue number2
DOIs
StatePublished - Mar 2007

Keywords

  • Automorphism group
  • Cycle decomposition
  • Edge-transitive graphs
  • Graph
  • Linking ring structure
  • Locally arc-transitive graph
  • Semisymmetric graph
  • Symmetry
  • Tetravalent graphs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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