Graphs and Combinatorics

, Volume 29, Issue 6, pp 1721–1731 | Cite as

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, New York (2008)Google Scholar
  2. 2.
    Cai L., Corneil D.: On cycle double covers of line graphs. Discrete Math. 102, 103–106 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Catlin P.A.: A reduction method to find spanning eulerian subgraphs. J. Graph Theory 12, 29–45 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Catlin P.A., Han Z., Lai H.-J.: Graphs without spanning eulerian subgraphs. Discrete Math. 160, 81–91 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Catlin, P.A., Lai, H.-J.: Spanning trails joining two given edges. In: Alavi, Y., Chartrand, G., Oellermann, O., Schwenk, A. (eds.) Graph Theory, Combinatorics, and Applications, vol. 1, pp. 207–222, Kalamazoo (1991)Google Scholar
  6. 6.
    Chen Z.-H., Lai H.-J., Lai H.Y.: Nowhere zero flows in line graph. Discrete Math. 230, 133–141 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, Y., Lai, H.-J., Li, H., Li, P.: Supereulerian graphs with width s and s-collapsible graphs (2012, submitted)Google Scholar
  8. 8.
    Gould R.: Advances on the Hamiltonian problem—a survey. Graphs Combin. 19, 7–52 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gu, X., Lai, H.-J., Yao, S.: Characterizations of minimal graphs with equal edge connectivity and spanning tree packing number (submitted)Google Scholar
  10. 10.
    Harary F., Nash-Williams C.St.J.A.: On eulerian and hamiltonian graphs and line graphs. Can. Math. Bull. 8, 701–709 (1965)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hsu, L.-H., Lin, C.-K.: Graph Theory and Interconnection Networks. CRC Press, Boca Raton (2009).Google Scholar
  12. 12.
    Huang P., Hsu L.: The spanning connectivity of the line graphs. Appl. Math. Lett. 24(9), 1614–1617 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jaeger, F.: Nowhere-zero flow problems. In: Beineke, L.W., Wilson, R.J. (eds.) Topics in Graph Theory, vol. 3, pp. 70–95. Academic Press, London (1988)Google Scholar
  14. 14.
    Lai H.-J., Li P., Liang Y., Xu J.: Reinforcing a matroid to have k disjoint bases. Appl. Math. 1, 244–249 (2010)CrossRefGoogle Scholar
  15. 15.
    Li, P.: Bases and cycles in matroids and graphs. Ph. D. Dissertation, West Virginia University (2012)Google Scholar
  16. 16.
    Liu D., Lai H.-J., Chen Z.-H.: Reinforcing the number of disjoint spanning trees. Ars Comb. 93, 113–127 (2009)MathSciNetMATHGoogle Scholar
  17. 17.
    Nash-Williams C.St.J.A.: Edge-disjoint spanning trees of finite graphs. J. Lond. Math. Soc. 36, 445–450 (1961)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Seymour, P.D.: Sums and circuits. In: Bondy, J.A., Murty, U.S.R. (eds.) Graph Theory and Related Topics, pp. 342–355. Academic Press, New York (1979)Google Scholar
  19. 19.
    Shao, Y.: Claw-free graphs and line graphs. Ph. D. Dissertation, West Virginia University (2005)Google Scholar
  20. 20.
    Szekeres G.: Polyhedral decompositions of cubic graphs. Bull. Aust. Math. Soc. 8, 367–387 (1973)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Thomassen C.: Reflections on graph theory. J. Graph Theory 10, 309–324 (1986)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Tutte W.T.: On the imbedding of linear graphs into surfaces. Proc. Lond. Math. Soc. Ser. 2(51), 464–483 (1949)Google Scholar
  23. 23.
    Tutte W.T.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 36, 221–230 (1961)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zhan S.M.: Hamiltonian connectedness of line graphs. Ars Comb. 22, 89–95 (1986)MATHGoogle Scholar

Copyright information

© Springer Japan 2012

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

Personalised recommendations