Classification of Symmetric Tabačjn Graphs

Abstract

A bicirculant is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we introduce the Tabaă?jn graphs, a family of pentavalent bicirculants which are a natural generalization of generalized Petersen graphs obtained from them by adding two additional perfect matchings between the two orbits of a semiregular automorphism. The main result is the classification of symmetric Tabaă?jn graphs. In particular, it is shown that the only such graphs are the complete graph $$K_{6}$$K6, the complete bipartite graph minus a perfect matching $$K_{6,6}-6K_2$$K6,6-6K2 and the icosahedron graph.