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Constructing an Infinite Family of Cubic 1-Regular Graphs

✍ Scribed by Yan-Quan Feng; Jin Ho Kwak

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 89 KB
Volume: 23
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.2002.0589

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✦ Synopsis

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.

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