Graphs and Combinatorics

, Volume 29, Issue 6, pp 1721–1731

# On Spanning Disjoint Paths in Line Graphs

• Ye Chen
• Zhi-Hong Chen
• Hong-Jian Lai
• Ping Li
• Erling Wei
Original Paper

## 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 $${u, v \in V(G)}$$ 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 $${u, v \in V(G)}$$ 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 $${u, v \in V(G)}$$ 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}.

### Keywords

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

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## Authors and Affiliations

• Ye Chen
• 1
• Zhi-Hong Chen
• 2
• Hong-Jian Lai
• 1
• 3
• Ping Li
• 4
• Erling Wei
• 5
1. 1.Department of MathematicsWest Virginia UniversityMorgantownUSA
2. 2.Department of Computer ScienceButler UniversityIndianapolisUSA
3. 3.College of Mathematics and System SciencesXinjiang UniversityUrumqiPeople’s Republic of China
4. 4.Department of MathematicsBeijing Jiaotong UniversityBeijingPeople’s Republic of China
5. 5.Department of MathematicsRenming University of ChinaBeijingPeople’s Republic of China