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 $2^m \equiv 1 \mod n$. In each case, $B(k,m,n)$ is a $2k$-valent vertex- and edge-transitive graph of order $mn^{k-1}$, and Bouwer showed that $B(k,6,9)$ is half-arc-transitive for all $k > 1$. For almost 45 years the question of exactly which of Bouwer's graphs are half-arc-transitive and which are arc-transitive has remained open, despite many attempts to answer it. In this paper, we use a cycle-counting argument to prove that almost all of the graphs constructed by Bouwer are half-arc-transitive. In fact, we prove that $B(k,m,n)$ is arc-transitive only when $n = 3$, or $(k,n) = (2,5)$, % and $m$ is a multiple of $4$, 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 $n > 5$. This gives an easy way to prove that there are infinitely many half-arc-transitive graphs of each even valency $2k > 2$.Comment: 16 pages, 1 figur

Lifting a prescribed group of automorphisms of graphs

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let $P$ be the Petersen graph, say, and let $\wp:\tilde{P}\to P$ be a regular covering projection. With the current covering machinery, it is straightforward to find $\wp$ with the property that every subgroup of \Aut(P) lifts via $\wp$. However, for constructing peculiar examples and in applications, this is usually not enough. Sometimes it is important, given a subgroup $G$ of \Aut(P), to find $\wp$ along which $G$ lifts but no further automorphism of $P$ does. For instance, in this concrete example, it is interesting to find a covering of the Petersen graph lifting the alternating group $A_5$ but not the whole symmetric group $S_5$. (Recall that \Aut(P)\cong S_5.) Some other time it is important, given a subgroup $G$ of \Aut(P), to find $\wp$ with the property that \Aut(\tilde{P}) is the lift of $G$. Typically, it is desirable to find $\wp$ satisfying both conditions. In a very broad sense, this might remind wallpaper patterns on surfaces: the group of symmetries of the dodecahedron is $S_5$, and there is a nice colouring of the dodecahedron (found also by Escher) whose group of symmetries is just $A_5$. In this paper, we address this problem in full generality.Comment: 10 page

The Separated Box Product of Two Digraphs

A new product construction of graphs and digraphs, based on the standard box product of graphs and called the separated box product, is presented, and several of its properties are discussed. Questions about the symmetries of the product and their relations to symmetries of the factor graphs are considered. An application of this construction to the case of tetravalent edge-transitive graphs is discussed in detail