Graphs and Combinatorics

, Volume 30, Issue 2, pp 501–510

# Collapsible Graphs and Hamiltonicity of Line Graphs

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

## Abstract

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.

## Keywords

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

## References

1. 1.
Bondy J.A., Bondy J.A.: Graph theory with application. Macmillan, London (1976)Google Scholar
2. 2.
Catlin P.A.: A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12, 29–45 (1988)
3. 3.
Catlin P.A., Han Z., Lai H.-J.: Graphs without spanning eulerian subgraphs. Discrete Math. 160, 81–91 (1996)
4. 4.
Chen, Z.-H., Lai, H.-J.: Reduction techniques for supereulerian graphs and related topics: an update. In: Tung-Hsin, K. (ed.) Combinatorics and Graph Theory, vol. 95, pp. 53–69. World Scientific, Singapore (1995)Google Scholar
5. 5.
Chen, Z.-H., Lai, H.-Y., Lai, H.-J., Weng, G.Q.: Jacson’s conjecture on wularian subgraphs. In: Alavi, Y., Lick, D.R., Liu, J. (eds.) Combinatorics, Graph Theory, Algorithms and Applications, pp. 53–58. The Proceeding of the Third China-USA Conference, Beijing, P.R. China (1993)Google Scholar
6. 6.
Esfahanian A.H., Hakimi S.L.: On computing a conditional edge-connectivity of a graph. Inf. Process. Lett. 27, 195–199 (1988)
7. 7.
Hu Z., Tian F., Wei B.: Hamilton connectivity of line graphs and claw-free graphs. J. Graph Theory 50, 130–141 (2005)
8. 8.
Hu, Z., Tian, F., Wei, B.: Some results on paths and cycles in claw-free graphs. PreprintGoogle Scholar
9. 9.
Kaiser, T., Vrána, P.: Hamilton cycles in 5-connected line graphs. Submitted on 20 Sep 2010. arXiv:1009.3754v1Google Scholar
10. 10.
Lai, H.-J., Shao, Y., Wu, H., Zhou J.: Every 3-connected, essentially 11-connected line graph is Hamiltonian. J. Combin. Theory Ser. B 96, 571–576 (2006)Google Scholar
11. 11.
Lai H.-J., Shao Y., Yu G., Zhan M.: Hamiltonian connectedness in 3-connected line graphs. Discrete Appl. Math. 157(5), 982–990 (2009)
12. 12.
Li, H.: A note on Hamiltonian claw-free graphs. Rapport de recherche no. 1022, LRI, UMR 8623 CNRS-UPS, Bat. 490, Université de Paris sud, 91405 Orsay, France (1996)Google Scholar
13. 13.
Matthews M.M., Sumner D.P.: Hamiltonian results in K 1, 3-free graphs. J. Graph Theory 8, 139–146 (1984)
14. 14.
Ryjáček Z.: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70, 217–224 (1997)
15. 15.
Shao, Y.: Claw-free graphs and line graphs. Ph.D dissertation, West Virginia University (2005)Google Scholar
16. 16.
Thomassen C.: Reflections on graph theory. J. Graph Theory 10, 309–324 (1986)
17. 17.
Yang, W., Lai, H., Li, H., Xiao, G.: Collapsible graphs and hamiltonicity of line graphs. SubmittedGoogle Scholar
18. 18.
Zhan S.: On Hamiltonian line graphs and connectivity. Discrete Math. 89, 89–95 (1991)
19. 19.
Zhan M.: Hamiltonicity of 6-connected line graphs. Discrete Appl. Math. 158, 1971–1975 (2010)

## 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