Hamiltonian connectedness in 3-connected line graphs

  • Authors:

  • Hong-Jian Lai;Yehong Shao;Gexin Yu;Mingquan Zhan

  • Affiliations:

  • Department of Mathematics, West Virginia University, Morgantown, WV 26506, United States;Arts and Science, Ohio University Southern, Ironton, OH 45638, United States;Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States;Department of Mathematics, Millersville University, Millersville, PA 17551, United States

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell's recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K"5-E(C"4), where C"4 is an cycle of length 4 in K"5) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if @k(L(G))=3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.