Arjana Žitnik: Half-arc-transitive graphs of arbitrary even valency greater than 2
A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than 2. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as B(k, m, n), defined for every triple (k, m, n) of integers greater than 1 with 2m ≡ 1 (mod n). In each case, B(k, m, n) is a 2k-valent vertex- and edge-transitive graph of order mnk-1, and Bouwer showed that B(k, 6, 9) is half-arc-transitive for all k > 1.
I will present a proof that almost all of the graphs constructed by Bouwer are half-arc-transitive. In fact, B(k, m, n) is arc-transitive only when n = 3, or (k, n) = (2, 5), or (k, m, n) = (2, 3, 7) or (2, 6, 7) or (2, 6, 21). In particular, B(k, m, n) is half-arc-transitive whenever m > 6 and n > 5, and hence there are infinitely many half-arc-transitive Bouwer graphs of each even valency 2k > 2.
This is joint work with Marston Conder.