Diameter and connectivity of (D; g)-cages

A (D; g)-cage is a graph having degree set D, girth g, and the minimum possible number of vertices. When D={r} the corresponding ({r}; g)-cage is clearly r-regular, and is called an (r; g)-cage. In this work we establish some structural properties of (D; g)-cages, generalizing some known properties for (r; g)-cages. In particular we prove that the diameter of a ({r, r+1}; g)-cage is at most g and also we prove that ({r, m}; g)-cages are 2-connected.

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

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

[3]  Camino Balbuena,et al.  Constructions of bi-regular cages , 2009, Discret. Math..

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

[5]  Camino Balbuena,et al.  On the order of ({r, m};g)-cages of even girth , 2008, Discret. Math..

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

[7]  Camino Balbuena,et al.  Superconnected digraphs and graphs with small conditional diameters , 2002, Networks.

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

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

[10]  Ping Wang,et al.  On cages with given degree sets , 1992, Discret. Math..

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

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

[13]  C. Balbuena,et al.  ( ,g )-cages with g10 are 4-connected , 2005 .

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

[15]  Noga Alon,et al.  The Moore Bound for Irregular Graphs , 2002, Graphs Comb..

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

[17]  Gary Chartrand,et al.  Graphs with prescribed degree sets and girth , 1981 .

[18]  Camino Balbuena,et al.  Monotonicity of the order of (D;g)-cages , 2011, Appl. Math. Lett..

[19]  Dhruv Mubayi,et al.  Connectivity and separating sets of cages , 1998, J. Graph Theory.

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

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

[22]  Weifa Liang,et al.  The minimum number of vertices with girth 6 and degree set D={r, m} , 2003, Discret. Math..

[23]  Saumyendra Sengupta,et al.  Graphs and Digraphs , 1994 .

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

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

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

[27]  S. F. Kapoor,et al.  Degree sets for graphs , 1977 .

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