All (k;g)-cages are edge-superconnected

A (k ;g)-cage is a k -regular graph with girth g and with the least possible number of vertices. In this article we prove that (k ;g)-cages are edge-superconnected if g is even. Earlier, Marcote and Balbuena proved that (k ;g)-cages are edge-superconnected if g is odd [Networks 43 (2004), 54–59]. Combining our results, we conclude that all (k ;g)cages are edge-superconnected. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(2), 102–11

[1]  H. Sachs,et al.  Regukre Graphen gegebener Taillenweite mit minimaler Knotenzahl , 1963 .

[2]  Norman Biggs,et al.  Constructions for Cubic Graphs with Large Girth , 1998, Electron. J. Comb..

[3]  Camino Balbuena,et al.  Edge-superconnectivity of cages , 2004 .

[4]  Pak-Ken Wong,et al.  Cages - a survey , 1982, J. Graph Theory.

[5]  Claudine Peyrat,et al.  Sufficient conditions for maximally connected dense graphs , 1987, Discret. Math..

[6]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[7]  W. T. Tutte.  A family of cubical graphs , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  Miguel Angel Fiol,et al.  Maximally connected digraphs , 1989, J. Graph Theory.

[9]  Miguel Angel Fiol,et al.  On the order and size of s-geodetic digraphs with given connectivity , 1997, Discret. Math..

[10]  Mirka Miller,et al.  All (k;g)-cages are k-edge-connected , 2005, J. Graph Theory.

[11]  Miguel Angel Fiol,et al.  Bipartite Graphs and Digraphs with Maximum Connectivity , 1996, Discret. Appl. Math..

[12]  Alexandru T. Balaban.  A trivalent graph of girth ten , 1972 .

[13]  Brendan D. McKay,et al.  The Smallest Cubic Graphs of Girth Nine , 1995, Combinatorics, Probability and Computing.

[14]  Ralph Tindell,et al.  Circulants and their connectivities , 1984, J. Graph Theory.

[15]  Dhruv Mubayi,et al.  Connectivity and separating sets of cages , 1998 .

[16]  Dragan Marusic,et al.  The 10-cages and derived configurations , 2004, Discret. Math..

[17]  Ping Wang,et al.  A Note on the Edge-Connectivity of Cages , 2003, Electron. J. Comb..

[18]  P. K. Wong.  On the smallest graphs of girth 10 and valency 3 , 1983, Discret. Math..

[19]  Terunao Soneoka,et al.  Super Edge-Connectivity of Dense Digraphs and Graphs , 1992, Discret. Appl. Math..

[20]  Miguel Angel Fiol,et al.  On the connectivity and the conditional diameter of graphs and digraphs , 1996, Networks.

[21]  F. Boesch.  Synthesis of reliable networks - a survey , 1986, IEEE Transactions on Reliability.

[22]  Hung-Lin Fu,et al.  Connectivity of cages , 1997, J. Graph Theory.

[23]  Markus Meringer,et al.  Fast generation of regular graphs and construction of cages , 1999, J. Graph Theory.

[24]  Christopher A. Rodger,et al.  (k, G)-cages Are 3-connected , 1999, Discret. Math..

[25]  Miguel Angel Fiol,et al.  Superconnectivity of bipartite digraphs and graphs , 1999, Discret. Math..