We investigate the existence of Hamilton paths in connected Cayley graphs on generalized dihedral groups. In particular, we show that a connected Cayley graph of valency at least three on a generalized dihedral group, whose order is divisible by four, is Hamiltonconnected, unless it is bipartite, in which case it is Hamilton-laceable.
Given two 2-regular graphs F 1 and F 2 , both of order n, the Hamilton-Waterloo Problem for F 1 and F 2 asks for a factorisation of the complete graph K n into α 1 copies of F 1 , α 2 copies of F 2 , and a 1-factor if n is even, for all non-negative integers α 1 and α 2 satisfying α 1 + α 2 = n−1 2 . We settle the Hamilton-Waterloo problem for all bipartite 2-regular graphs F 1 and F 2 where F 1 can be obtained from F 2 by replacing each cycle with a bipartite 2-regular graph of the same order.
Abstract:The circulant G = C(n, S), where S ⊆ Z n \ {0}, is the graph with vertex set Z n and edge set E(G) = {{x, x + s}|x ∈ Z n , s ∈ S}. It is shown that for n odd, every 6-regular connected circulant C(n, S) is decomposable into Hamilton cycles.
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