Spanning 3-connected index of graphs

Wei Xiong, Zhao Zhang () and Hong-Jian Lai
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Wei Xiong: Xinjiang University
Zhao Zhang: Xinjiang University
Hong-Jian Lai: Xinjiang University

Journal of Combinatorial Optimization, 2014, vol. 27, issue 1, No 16, 199-208

Abstract: 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.

Keywords: Spanning k-connected index; 3-triangular graph; Line graph; 2-collapsible (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-012-9583-7

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