Constructing an Infinite Family of Cubic 1-Regular Graphs

Abstract

A graph is 1-regular if its automorphism group acts regularly on the set of its arcs. Miller [J. Comb. Theory, B, 10 (1971), 163-182] constructed an infinite family of cubic 1-regular graphs of order 2 p, where p ≥ 13 is a prime congruent to 1 modulo 3. Marušič and Xu [J. Graph Theory, 25 (1997), 133-138] found a relation between cubic 1-regular graphs and tetravalent half-transitive graphs with girth 3 and Alspach et al. [J. Aust. Math. Soc. A, 56 (1994), 391-402] constructed infinitely many tetravalent half-transitive graphs with girth 3. Using these results, Miller's construction can be generalized to an infinite family of cubic 1-regular graphs of order 2n, where n ≥ 13 is odd such that 3 divides ϕ(n), the Euler function of n. In this paper, we construct an infinite family of cubic 1-regular graphs with order 8(k 2 + k + 1)(k ≥ 2) as cyclic-coverings of the three-dimensional Hypercube.

Cubic symmetric graphs of order a small number times a prime or a prime square

“…It may be shown that all cubic 1-regular Cayley graphs on the dihedral groups are exactly those graphs generalized by Miller's construction. Recently, more 1-regular cubic graphs were constructed by the authors [15][16][17]. Also, as shown in [40] or [41], one can see an importance of a study for cubic 1-regular graphs in connection with chiral (that is regular and irreflexible) maps on a surface by means of tetravalent half-transitive graphs or in connection with symmetries of hexagonal molecular graphs on the torus.…”

Bridging Semisymmetric and Half-Arc-Transitive Actions on Graphs

“…The first one is alter-incomplete with alter-sequence (2, 4), whereas the second one is alter-complete with alter-sequence (3,12). As in the previous two examples F(L(Q 3 ); H 1 ) is disconnected and not semisymmetric.…”

“…The line graph of the graph F56A in row 28 in Table 1 is an example of such a graph with alter-exponent 2. In fact, F56A is the smallest member of an infinite family of 1-regular Z k 2 +k+1 -covers of Q 3 (given in [12]), for which the corresponding line graphs are 1 2 -arc-transitive of alter-exponent 2. It may be easily seen that these graphs are alter-incomplete with alter-sequence [2(k 2 + k + 1), 4(k 2 + k + 1)].…”

On 1-Arc-regular Graphs

“…However, all known examples of 1-arc-regular trivalent graphs are Cayley graphs and have soluble automorphism groups. Moreover it was shown in [7] In this paper, we first discuss question (a) and prove that if G is soluble then must be a Cayley graph of a subgroup of G. Then we turn to questions (b) and (c) by considering 1-arc-regular trivalent Cayley graphs of a finite nonabelian simple group. We give a sufficient condition under which one can guarantee that Cay(G, S) is 1-arc-regular.…”

“…In the literature, the first example of 1-arc-regular trivalent graphs was given by Frucht [4] in 1952. Much later, some other examples were given in [3,13], and very recently more examples were obtained in [7][8][9]. However, all known examples of 1-arc-regular trivalent graphs are Cayley graphs and have soluble automorphism groups.…”