Connectivity of graphs with given girth pair

Abstract

Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209-218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h) such that g is odd and h g + 3 is even has high (vertex-)connectivity if its diameter is at most h − 3. The edge version of all results is also studied.

Superconnectivity of graphs with odd girth g and even girth h

“…Item (i) of Theorem 1 was improved for graphs with girth pair (g, h), odd g and even h ≥ g + 3 in [3].…”

On local connectivity of graphs with given clique number

“…We call a graph G maximally connected when (G) = (G) and maximally local connected when (u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. Because of (G)≤ (G), there exists a special interest on graphs G with (G)= (G). Different authors have presented sufficient conditions for graphs to be maximally connected, as, for example, Balbuena et al [1], Esfahanian [2], Fàbrega and Fiol [3,4], Fiol [5], Hellwig and Volkmann [6], Soneoka et al [9] and Topp and Volkmann [10]. For more information on this topic we refer the reader to the survey article by Hellwig and Volkmann [7].…”

Handbook of Graph Theory

“…[BaCeDiGVMa07,BaGVMo11] Let G be a graph with minimum degree 3, diameter D, girth pair (g, h), odd g and even h with g + 3  h < 1, and connectivities  and . (a) If D  h 3, then = .…”