### The arc-types of Cayley graphs

#### Abstract

Let *X* be a finite vertex-transitive graph of valency *d*, and let *A* be the full automorphism group of *X*. Then the *arc-type* of *X* is defined in terms of the sizes of the orbits of the action of the stabiliser *A*_{v} of a given vertex *v* on the set of arcs incident with *v*. Specifically, the arc-type is the partition of *d* as the sum *n*_{1} + *n*_{2} + … + *n*_{t} + (*m*_{1} + *m*_{1}) + (*m*_{2} + *m*_{2}) + … + (*m*_{s} + *m*_{s}), where *n*_{1}, *n*_{2}, …, *n*_{t} are the sizes of the self-paired orbits, and *m*_{1}, *m*_{1}, *m*_{2}, *m*_{2}, …, *m*_{s}, *m*_{s} are the sizes of the non-self-paired orbits, in descending order.

In a recent paper, it was shown by Conder, Pisanski and Žitnik that with the exception of the partitions 1 + 1 and (1 + 1) for valency 2, every such partition occurs as the arc-type of some vertex-transitive graph. In this paper, we extend this to show that in fact every partition other than 1, 1 + 1 and (1 + 1) occurs as the arc-type of infinitely many connected finite Cayley graphs with the given valency *d*. As a consequence, this also shows that for every *d* > 2, there are infinitely many finite zero-symmetric graphs (or GRRs) of valency *d*.

#### Keywords

DOI: https://doi.org/10.26493/1855-3974.1327.6ee

ISSN: 1855-3974

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