Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m, n)-semiregular for some integers m ≥ 1, n ≥ 2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ m has at least one fixed vertex. A half-arc-transitive graph is a vertex-and edge-but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic halfarc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.Keywords: Graph; Metacirculant graph; Half-arc-transitive; Tightly attached; Automorphism group 1 Introductory and historic remarks Throughout this paper graphs are assumed to be finite and, unless stated otherwise, simple, connected and undirected (but with an implicit orientation of the edges when appropriate). For group-theoretic concepts not defined here we refer the reader to [4,9,32], and for graph-theoretic terms not defined here we refer the reader to [5].Given a graph X we let V (X), E(X), A(X) and AutX be the vertex set, the edge set, the arc set and the automorphism group of X, respectively. A graph X is said to be vertex-transitive, edge-transitive and arc-transitive if its automorphism group AutX acts transitively on V (X), E(X) and A(X), respectively. We say that X is half-arctransitive provided it is vertex-and edge-but not arc-transitive. More generally, by a half arc-transitive action of a subgroup G ≤ AutX on X we mean a vertex-and edge-but not arc-transitive action of G on X. In this case we say that the graph X is (G, 1 2 )-arc-transitive, and we say that the graph X is (G, 1 2 , H)-arc-transitive when it needs to be stressed that the vertex stabilizers G v (for v ∈ V (X)) are isomorphic to a particular subgroup H ≤ G. By a classical result of Tutte [30, 7.35, p.59], a graph admitting a half-arc-transitive group action is necessarily of even valency. A 1 Supported in part by "ARRS -Agencija za znanost Republike Slovenije", program no. P1-0285.

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We prove that if Cay(G; S) is a connected Cayley graph with n vertices, and the prime factorization of n is very small, then Cay(G; S) has a hamiltonian cycle. More precisely, if p, q, and r are distinct primes, then n can be of the form kp with 24 = k < 32, or of the form kpq with k ≤ 5, or of the form pqr, or of the form kp 2 with k ≤ 4, or of the form kp 3 with k ≤ 2.

We present some new discoveries about graphs that are half-arc-transitive (that is, vertex-and edge-transitive but not arc-transitive). These include the recent discovery of the smallest half-arc-transitive 4-valent graph with vertex-stabiliser of order 4, and the smallest * The first author was supported by a James Cook Fellowship and a Marsden Fund grant (UOA1015) from the Royal Society of New Zealand with vertex-stabiliser of order 8, two new half-arc-transitive 4-valent graphs with dihedral vertex-stabiliser D 4 (of order 8), and the first known half-arc-transitive 4-valent graph with vertex-stabiliser that is neither abelian nor dihedral. We also use half-arc-transitive group actions to provide an answer to a recent question of Delorme about 2-arc-transitive digraphs that are not isomorphic to their reverse.