Cycles in Cayley graphs of order 2p with a proscribed vertex (Q1086248)

A graph G has the property C(m,n) if, whenever M, N are disjoint sets of vertices of G with \(| M| =m\) and \(| N| =n\), there exists a cycle in G which includes all the vertices of M and which avoids all vertices of N. Let G be a group generated by a subset X of G. We denote by G(X) the graph whose vertices are elements of G and two vertices a,b are adjacent if and only if \(a^{-1}b\) is in \(X\cup X^{-1}\). The graph G(X) is known as a Cayley graph. For a group G of order 2p where p is a prime, G(X) has the property C(2p-1,1) if and only if X contains an element of order p. This, in turn, is equivalent to G(X) being non- bipartite; or G(X) being Hamilton-connected. If the order of each element in X is 2, then G(X) does not have the property C(p,1). Sufficient conditions will be given to ensure that G(X) has the property C(p-1,1).