Hamiltonian paths and hamiltonian connectivity in graphs

AbstractLet G be a 2-connected graph with n vertices such that d(u)+d(v)+d(w)-|N(u)∩N(v)∩N(w)| ⩾n + 1 holds for any triple of independent vertices u, v and w. Then for any distinct vertices u and v such that {u, v} is not a cut vertex set of G, there is a hamiltonian path between u and v. In particular, if G is 3-connected, then G is hamiltonian-connected. This is closely related to the main result in Flandrin et al. (1991) and generalizes a theorem of Ore (1963) and a theorem of Faudree et al. (1989)

On Hamiltonicity of {claw, net}-free graphs

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with given two different vertices s, t and an edge e to have (1)a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian s- and st-paths containing edge e when G has connectivity one, and (4) a Hamiltonian cycle containing e when G is 2-connected. These results imply that a connected {claw, net}-free graph has a Hamiltonian path and a 2-connected {claw, net}-free graph has a Hamiltonian cycle [D. Duffus, R.J. Gould, M.S. Jacobson, Forbidden Subgraphs and the Hamiltonian Theme, in The Theory and Application of Graphs (Kalamazoo, Mich., 1980$), Wiley, New York (1981) 297--316.] Our proofs of (1)-(4) are shorter than the proofs of their corollaries in [D. Duffus, R.J. Gould, M.S. Jacobson] and provide polynomial-time algorithms for solving the corresponding Hamiltonicity problems. Keywords: graph, claw, net, {claw, net}-free graph, Hamiltonian path, Hamiltonian cycle, polynomial-time algorithm.Comment: 9 page

Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.

We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time