http://nova.newcastle.edu.au/vital/access/services/Feed ${session.getAttribute("locale")} 5 http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:11492 A semiregular cage is a graph with degree set {r,r +1}, girth g and the smallest possible order. In this work we prove that semiregular cages with r ≥ 3 and odd girth g ≥ 7 are t-connected with t ≥ |√ r + (3/2)² - 1/2|. 2012-09-10T05:00:29.049Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9940 The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 (1983), 209–218]. A (δ, g)-cage is a smallest δ-regular graph with girth g. For all δ ≥ 3 and odd girth g ≥ 5, Harary and Kovács conjectured the existence of a (δ,g)-cage that contains a cycle of length g + 1. In the main theorem of this article we present a lower bound on the order of a δ-regular graph with odd girth g ≥ 5 and even girth h ≥ g + 3. We use this bound to show that every (δ,g)-cage with δ ≥ 3 and g ∈ {5,7} contains a cycle of length g + 1, a result that can be seen as an extension of the aforementioned conjecture by Harary and Kovács for these values of δ, g. Moreover, for every odd g ≥ 5, we prove that the even girth of all (δ,g)-cages with δ large enough is at most (3g − 3)/2. 2012-02-08T22:40:06.378Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:9351 In this paper, first we prove that any graph G is 2-connected if diam(G)≤g−1 for even girth g, and for odd girth g and maximum degree Δ≤2δ−1 where δ is the minimum degree. Moreover, we prove that any graph G of diameter diam(G)≤g−2 satisfies that (i) G is 5-connected for even girth g and Δ≤2δ−5, and (ii) G is super-κ for odd girth g and Δ≤3δ/2−1. 2011-11-13T23:00:13.543Z ]]>