On prime-valent symmetric bicirculants and Cayley snarks

Abstract

An n-bicirculant (in short, a bicirculant) is a graph admitting a non-identity automorphism having two cycles of equal length n in its cycle decomposition (called a (2,n)-semiregular automorphism). A graph is said to be symmetric if its automorphism group acts transitively on the set of its arcs. In this paper it is shown that a connected bicirculant X ≠ K4 of prime valency admitting a group of automorphisms containing a (2,n)-semiregular automorphism and acting regularly on the set of arcs is near-bipartite (that is, with the chromatic number at most 3). Combining this result with the theory of Cayley maps new partial results are obtained in regards to the well-known conjecture that there are no snarks amongst Cayley graphs. © 2013 Springer-Verlag.