Abstract
A graph Γ has the property C(m,n) if, whenever M, N are disjoint sets of vertices of Γ with |M|=m and |N|=n, there exists a cycle in Γ which includes all the vertices of M and which avoids all the 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 e X U X−1. The graph G(X) is known as a Cayley graph. For an abelian group G of order p, G(X) has the property C(p−1,1) if and only if G(X) is neither cyclic nor bipartite, which in turn, is equivalent to G(X) being hamilton-connected. Moreover, if G(X) is bipartite of order 2t but not cyclic, then G(X) has the property C(t−1,1) but not C(t,1). If G(X) is cyclic then it is not C(m,1) for any m.
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References
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© 1984 Springer-Verlag
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Chen, C.C., Holton, D.A. (1984). Cycles in abelian cayley graphs with a proscribed vertex. In: Koh, K.M., Yap, H.P. (eds) Graph Theory Singapore 1983. Lecture Notes in Mathematics, vol 1073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073102
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DOI: https://doi.org/10.1007/BFb0073102
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