Abstract :
In 1991, Alspach, Marušič, and Nowitz proved that there are infinitely many 12-transitive graphs of degree 4. Their graphs were found among metacirculants M(α; m, n), which have vertex set {vji: i ϵ Zm, j ϵ Zn} and edge set {vji vj+δαii+1: i ϵ Zn, δ ϵ {− 1, 1}} with the additional condition that α ∈ Zn∗ has order m or 2m. Examining only the cases when both m and n are odd, they showed that the graphs M(α; 3, n) are 12-transitive when n ⩾ 9 and gave a sufficient condition for M(α; m, n) to be 12-transitive when m is composite and n is prime. In this paper, we give a simple generalization of this condition. We also show that the graphs M(α; 2, n) are arc-transitive. Then we examine the graphs M(α; 4, n). We prove that they are arc-transitive when the order of α is 4 with α2 ≡ −1 (mod n) and 12-transitive when either the order of α is 8 or the order of α is 4 with α2 ≢ −1 (mod n) and n is not a multiple of 4.
