# On Spanning Disjoint Paths in Line Graphs

Ye Chen, Zhi Hong Chen, Hong Jian Lai, Ping Li, Erling Wei

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

## Abstract

Spanning connectivity of graphs has been intensively investigated in the study of interconnection networks (Hsu and Lin, Graph Theory and Interconnection Networks, 2009). For a graph G and an integer s > 0 and for u, v ∈ V(G) with u ≠ v, an (s; u, v)-path-system of G is a subgraph H consisting of s internally disjoint (u,v)-paths. A graph G is spanning s-connected if for any u, v ∈ V(G) with u ≠ v, G has a spanning (s; u, v)-path-system. The spanning connectivityκ*(G) of a graph G is the largest integer s such that G has a spanning (k; u, v)-path-system, for any integer k with 1 ≤ k ≤ s, and for any u, v ∈ V(G) with u ≠ v. An edge counter-part of κ*(G), defined as the supereulerian width of a graph G, has been investigated in Chen et al. (Supereulerian graphs with width s and s-collapsible graphs, 2012). In Catlin and Lai (Graph Theory, Combinatorics, and Applications, vol. 1, pp. 207-222, 1991) proved that if a graph G has 2 edge-disjoint spanning trees, and if L(G) is the line graph of G, then κ*(L(G)) ≥ 2 if and only if κ(L(G)) ≥ 3. In this paper, we extend this result and prove that for any integer k ≥ 2, if G0, the core of G, has k edge-disjoint spanning trees, then κ*(L(G)) ≥ k if and only if κ(L(G)) ≥ max{3, k}.

Original language English (US) 1721-1731 11 Graphs and Combinatorics 29 6 https://doi.org/10.1007/s00373-012-1237-0 Published - Nov 2013 Yes

## Keywords

• Collapsible graphs
• Connectivity
• Hamiltonian linegraph
• Hamiltonian-connected line graph
• Spanning connectivity
• Supereulerian graphs

## ASJC Scopus subject areas

• Theoretical Computer Science
• Discrete Mathematics and Combinatorics

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