# Spanning 3-connected index of graphs

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## 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.

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## Acknowledgments

The work is supported by NSFC (61222201).

## Author information

Authors

### Corresponding author

Correspondence to Zhao Zhang.

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Xiong, W., Zhang, Z. & Lai, HJ. Spanning 3-connected index of graphs. J Comb Optim 27, 199–208 (2014). https://doi.org/10.1007/s10878-012-9583-7