Graphs and Combinatorics

, Volume 30, Issue 2, pp 501–510 | Cite as

Collapsible Graphs and Hamiltonicity of Line Graphs

  • Weihua YangEmail author
  • Hong-Jian Lai
  • Hao Li
  • Xiaofeng Guo
Original Paper


Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai (Combinatorics and Graph Theory, vol 95, World Scientific, Singapore, pp 53–69; Conjecture 8.6 of 1995) conjectured that every 3-edge connected and essentially 6-edge connected graph is collapsible. Denote D 3(G) the set of vertices of degree 3 of graph G. For \({e = uv \in E(G)}\), define d(e) = d(u) + d(v) − 2 the edge degree of e, and \({\xi(G) = \min\{d(e) : e \in E(G)\}}\). Denote by λ m (G) the m-restricted edge-connectivity of G. In this paper, we prove that a 3-edge-connected graph with \({\xi(G)\geq7}\), and \({\lambda^3(G)\geq7}\) is collapsible; a 3-edge-connected simple graph with \({\xi(G)\geq7}\), and \({\lambda^3(G)\geq6}\) is collapsible; a 3-edge-connected graph with \({\xi(G)\geq6}\), \({\lambda^2(G)\geq4}\), and \({\lambda^3(G)\geq6}\) with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with \({\xi(G)\geq6}\), and \({\lambda^3(G)\geq5}\) with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with \({\xi(G)\geq5}\), and \({\lambda^2(G)\geq4}\) with at most 9 vertices of degree 3 is collapsible. As a corollary, we show that a 4-connected line graph L(G) with minimum degree at least 5 and \({|D_3(G)|\leq9}\) is Hamiltonian.


Thomassen’s conjecture Line graph Supereulerian graph Collapsible graph Hamiltonian graph Dominating eulerian subgraph 


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Copyright information

© Springer Japan 2013

Authors and Affiliations

  • Weihua Yang
    • 1
    • 2
    Email author
  • Hong-Jian Lai
    • 3
  • Hao Li
    • 2
  • Xiaofeng Guo
    • 4
  1. 1.Department of MathematicsTaiyuan University of TechnologyTaiyuanChina
  2. 2.Laboratoire de Recherche en InformatiqueC.N.R.S., University de Paris-sudOrsay cedexFrance
  3. 3.Department of MathematicsWest Virginia UniversityMorgantownUSA
  4. 4.School of Mathematical ScienceXiamen UniversityXiamen, FujianChina

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