TY - JOUR

T1 - Tetravalent edge-transitive graphs of girth at most 4

AU - Potočnik, Primož

AU - Wilson, Steve

N1 - Funding Information:
Keywords: Graph; Automorphism group; Symmetry; Edge-transitive graphs; Tetravalent graphs; Locally arc-transitive graph; Semisymmetric graph; Cycle decomposition; Linking ring structure E-mail addresses: potocnik@math.auckland.ac.nz, primoz.potocnik@fmf.uni-lj.si (P. Potocˇnik), stephen.wilson@nau.edu (S. Wilson). 1 Address for correspondence: Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia. 2 The author gratefully acknowledges the support of the US Department of State and the Fulbright Scholar Program who sponsored his visit to Northern Arizona University in spring 2004.

PY - 2007/3

Y1 - 2007/3

N2 - 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.

AB - 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.

KW - Automorphism group

KW - Cycle decomposition

KW - Edge-transitive graphs

KW - Graph

KW - Linking ring structure

KW - Locally arc-transitive graph

KW - Semisymmetric graph

KW - Symmetry

KW - Tetravalent graphs

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U2 - 10.1016/j.jctb.2006.03.007

DO - 10.1016/j.jctb.2006.03.007

M3 - Article

AN - SCOPUS:33846160597

SN - 0095-8956

VL - 97

SP - 217

EP - 236

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

IS - 2

ER -