Arc-regular cubic graphs of order four times an odd integer.

A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4 m where m is an odd integer. Using the Gorenstein-Walter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of automorphisms that is isomorphic to a subgroup of Aut(PSL(2, q)) containing PSL(2, q) for some odd prime-power q. Also a construction is given for infinitely many such graphs-namely a family of Cayley graphs for the groups PSL(2, p) where p is an odd prime; the smallest of these has order 9828. [ABSTRACT FROM AUTHOR]