Spanning 3-connected index of graphs

For an integer s > 0 and for u, v ∈ V (G) with u = v, an (s; u, v)-pathsystem 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 κ∗(G) of graph G is the largest integer s such that for any integer k with 1 ≤ k ≤ s and for any u, v ∈ V (G) with u = v, G has a spanning (k; u, v)-pathsystem. Let G be a simple connected graph that is not a path, a cycle or a K1,3. The spanning k-connected index of G, written sk(G), is the smallest nonnegative integer m such that Lm(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 K3}, where a divalent path in G is a path whose interval vertices have degree two in G. In this paper, we prove that s3(G) ≤ l(G) + 6. The key proof to this result is that every connected 3-triangular graph is 2-collapsible.