Spanning 3-connected index of graphs
For an integer \(s>0\) and for \(u,v\in V(G)\) with \(u\ne v\), an \((s;u,v)\)-path-system of G is a subgraph H of G consisting of s internally disjoint (u, v)-paths, and such an H is called a spanning \((s;u,v)\)-path system if \(V(H)=V(G)\). The spanning connectivity \(\kappa ^{*}(G)\) of graph G is the largest integer s such that for any integer k with \(1\le k \le s\) and for any \(u,v\in V(G)\) with \(u\ne v\), G has a spanning (\(k;u,v\))-path-system. Let G be a simple connected graph that is not a path, a cycle or a \(K_{1,3}\). The spanning k-connected index of G, written \(s_{k}(G)\), is the smallest nonnegative integer m such that \(L^m(G)\) is spanning k-connected. Let \(l(G)=\max \{m:\,G\) has a divalent path of length m that is not both of length 2 and in a \(K_{3}\)}, where a divalent path in G is a path whose interval vertices have degree two in G. In this paper, we prove that \(s_{3}(G)\le l(G)+6\). The key proof to this result is that every connected 3-triangular graph is 2-collapsible.
spanning k-connected index 3-triangular graph line graph 2-collapsible