Edge-transitive regular Zn-covers of the Heawood graph

AbstractA regular cover of a graph is said to be an edge-transitive cover if the fibre-preserving automorphism subgroup acts edge-transitively on the covering graph. In this paper we classify edge-transitive regular Zn-covers of the Heawood graph, and obtain a new infinite family of one-regular cubic graphs. Also, as an application of the classification of edge-transitive regular Zn-covers of the Heawood graph, we prove that any bipartite edge-transitive cubic graph of order 14p is isomorphic to a normal Cayley graph of dihedral group if the prime p>13

Cubic symmetric graphs of order a small number times a prime or a prime square

AbstractA graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K3,3 and the s-regular cyclic or elementary abelian coverings of the complete graph K4 for each s⩾1 are classified when the fibre-preserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regular graphs with girth 12 is found, in which the smallest one has order 2058. As an interesting application, a complete list of pairwise non-isomorphic s-regular cubic graphs of order 4p, 6p, 4p2 or 6p2 is given for each s⩾1 and each prime p

Classifying cubic symmetric graphs of order 8p or 8p2

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, we classify the s-regular elementary Abelian coverings of the three-dimensional hypercube for each s >= 1 whose fibre-preserving automorphism subgroups act arc-transitively. This gives a new infinite family of cubic 1-regular graphs, in which the smallest one has order 19208. As an application of the classification, all cubic symmetric graphs of order 8p or 8p(2) are classified for each prime p, as a continuation of the first two authors' work, in Y.-Q. Feng, J.H. Kwak [Cubic symmetric graphs of order a small number times a prime or a prime square (submitted for publication)] in which all cubic symmetric graphs of order 4p, 4p(2), 6p or 6p(2) are classified and of Cheng and Oxley's classification of symmetric graphs of order 2p, in Y. Cheng, J. Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42 (1987) 196-211]. (C) 2004 Elsevier Ltd. All rights reserved.X1170sciescopu