http://nova.newcastle.edu.au/vital/access/services/Feed ${session.getAttribute("locale")} 5 http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:12471 A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1)² ≤ δ + √ δ − 2 < r² and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying r(r−1)² / 4 +1+2r(r − 1) ≤ δ. These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected. 2013-01-23T01:00:03.697Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:10371 A (k,g)-cage is a k-regular graph with girth g that has the fewest number of vertices. It has been conjectured (Fu et al., J Graph Theory 24 (1997), 187–191) that all (k,g)-cages are k-connected for k ≥ 3. A connected graph G is said to be superconnected if every minimum cut-set S is the neighborhood of a vertex of minimum degree. Moreover, if G − S has precisely two components, then G is called tightly superconnected. It was shown (Xu et al., Ars Combin 64 (2002), 181–192) that every (4,g)-cage is 4-connected. In this article, we prove that every (4,g)-cage is tightly superconnected when g is even and g ≥ 12. 2012-03-12T03:50:08.122Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:5502 A (k,g)-cage is a k-regular graph with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)-cages for g even. 2012-01-30T04:04:28.862Z ]]>