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Tree Connectivities of Cayley Graphs on Abelian Groups with Small Degrees

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Abstract

The generalized k-connectivity \(\kappa _{k}(G)\) and the generalized k-edge-connectivity \(\lambda _k(G)\) of a graph G, also known as the tree connectivities, were introduced by Hager (J Comb Theory Ser B 38:179–189, 1985) and Li et al. (Discret Math Theor Comput Sci 14:43–54, 2012), respectively. In this paper, we study these invariants for Cayley graphs on Abelian groups with degree 3 or 4. When G is cubic, we prove \(\kappa _k(G) = \lambda _k(G) = 2\) for \(3\le k\le 6\) and \(\kappa _k(G) = \lambda _k(G) = 1\) for \(7\le k\le n\). When G has degree 4, we obtain \(\kappa _{3}(G)=\lambda _3(G)=3\), \(\lambda _k(G) = 2\) and \(\kappa _k(G)\le 2\) for \(k\ge 8\), and \(\kappa _k(G) = 2\) for \(k=n-1,n\).

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Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. Yuefang Sun was supported by NSFC No. 11401389. Sanming Zhou was supported by a Future Fellowship (FT110100629) of the Australian Research Council.

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Correspondence to Yuefang Sun.

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Communicated by Xueliang Li.

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Sun, Y., Zhou, S. Tree Connectivities of Cayley Graphs on Abelian Groups with Small Degrees. Bull. Malays. Math. Sci. Soc. 39, 1673–1685 (2016). https://doi.org/10.1007/s40840-015-0147-8

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  • DOI: https://doi.org/10.1007/s40840-015-0147-8

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