# Half-arc-transitive graphs of arbitrary even valency greater than 2 (Seminar DM)

Half-arc-transitive graphs of arbitrary even valency greater than 2

Arjana Žitnik

Torek, 23. decembra 2014, od 10h do 12h, Plemljev seminar, Jadranska 19

Povzetek: 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 $2^m \equiv 1 \pmod n$. 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.