Applied Mathematics Letters 25 (2012) 1676–1680 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml A new bound for the connectivity of cages C. Balbuena ∗ , J. Salas Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Campus Nord, Edifici C2, C/ Jordi Girona, 1-3, 08034 Barcelona, Spain article info Article history: Received 18 October 2010 Received in revised form 27 January 2012 Accepted 27 January 2012 Keywords: Cage Vertex-cut Vertex-connectivity Girth 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. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction We only consider undirected simple graphs without loops or multiple edges. Unless otherwise stated, we follow [1] for basic terminology and definitions. Let G stand for a graph with vertex set V = V (G) and edge set E = E (G). A graph G is called connected if every pair of vertices is joined by a path. A vertex cut in a graph G is a set X of vertices of G such that G − X is disconnected. Every graph G different from a complete graph has a vertex cut. The vertex connectivity κ(G) of a noncomplete graph G is the minimum cardinality of a vertex cut. A noncomplete graph is said to be k-connected if κ(G) ≥ k. The set of vertices adjacent to a vertex v is denoted by N (v), the degree of a vertex v is deg(v) =|N (v)|, and the minimum degree δ = δ(G) is the minimum degree over all vertices of G. A graph is called r -regular if every vertex of the graph has degree r . The length of a shortest cycle in a graph G is called the girth of G. An r -regular graph with girth g is called an (r , g )-graph. An (r , g )-graph is said to be an (r , g )-cage if it has the least possible number of vertices. Cages were introduced by Tutte [2] in 1947. In 1963, Erdös and Sachs [3] proved that (r , g )-cages exist for any given value of the pair (r , g ). Since then, a large amount of the research on cages has been devoted to their construction. For more information on this problem see the survey by Wong [4], or the survey by Holton and Sheehan [5], or the more recent one by Exoo and Jajcay [6]. A basic structural property of a graph is its connectivity. Concerning the connectivity of cages, Fu et al. proved that (r , g )-cages are 2-connected [7]. In addition they posed the following conjecture. Conjecture 1.1 ([7]). Every (r , g )-cage is r -connected. This conjecture is clearly true for g = 3, 4 because (r ; 3)-cages are complete graphs and (r ; 4)-cages are complete bipartite graphs. Further, this conjecture has been shown to hold for all r ≥ 3 where r − 1 is a prime power and g = 5, 6, 7, 8[8], and for g = 11, 12 [9], and also, when r = 3, 4 for all g ≥ 3[10,7,11–13]. Later, Lin et al. [14] proved that every (r , g )-cage with r ≥ 3 and odd girth g ≥ 7 is √ r + 1  -connected. Later Lin et al. [15] proved that every (r , g )-cage with r ≥ 3 and even girth g ≥ 6 is (t + 1)-connected, t being the largest integer such that t 3 + 2t 2 ≤ r . These results have recently been improved by Lu et al. [16]. These authors have proved that every (r , g )-cage with r ≥ 3 and odd girth g ≥ 9 ∗ Corresponding author. E-mail addresses: m.camino.balbuena@upc.edu (C. Balbuena), julian.salas@upc.edu (J. Salas). 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.036