A new bound for the connectivity of cages

Abstract An ( r , g ) -cage is an r -regular graph of girth g of minimum order. We prove that all ( r , g ) -cages are at least ⌈ r / 2 ⌉ -connected for every odd girth g ≥ 7 by means of a matrix technique which allows us to construct graphs without short cycles. This lower bound on the vertex connectivity of cages is a new advance in proving the conjecture of Fu, Huang and Rodger which states that all ( r , g ) -cages are r -connected.

[1]  Ping Wang,et al.  On the Connectivity of (4;g)-Cages , 2002, Ars Comb..

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

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

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

[5]  Camino Balbuena,et al.  (delta, G)-cages with G Geq 10 Are 4-connected , 2005, Discret. Math..

[6]  Juan José Montellano-Ballesteros,et al.  On upper bounds and connectivity of cages , 2007, Australas. J Comb..

[7]  Mehdi Behzad,et al.  Graphs and Digraphs , 1981, The Mathematical Gazette.

[8]  Qinglin Yu,et al.  New improvements on connectivity of cages , 2011 .

[9]  J. Fàbrega,et al.  On the superconnectivity and the conditional diameter of graphs and digraphs , 1999 .

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

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

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

[13]  G. Exoo,et al.  Dynamic Cage Survey , 2011 .

[14]  Derek Allan Holton,et al.  The Petersen graph , 1993, Australian mathematical society lecture series.

[15]  Mirka Miller,et al.  On the connectivity of (k, g)-cages of even girth , 2008, Discret. Math..

[16]  Camino Balbuena,et al.  On the connectivity of cages with girth five, six and eight , 2007, Discret. Math..

[17]  Mirka Miller,et al.  Improved lower bound for the vertex connectivity of (delta;;g)-cages , 2005, Discret. Math..

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

[19]  Hung-Lin Fu,et al.  Connectivity of cages , 1997 .