Spanning 3-connected index of graphs

Abstract

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.