329 research outputs found
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
Heavy subgraphs, stability and hamiltonicity
Let be a graph. Adopting the terminology of Broersma et al. and \v{C}ada,
respectively, we say that is 2-heavy if every induced claw () of
contains two end-vertices each one has degree at least ; and
is o-heavy if every induced claw of contains two end-vertices with degree
sum at least in . In this paper, we introduce a new concept, and
say that is \emph{-c-heavy} if for a given graph and every induced
subgraph of isomorphic to and every maximal clique of ,
every non-trivial component of contains a vertex of degree at least
in . In terms of this concept, our original motivation that a
theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and
-c-heavy graph is hamiltonian, where is the graph obtained from a
triangle by adding three disjoint pendant edges. In this paper, we will
characterize all connected graphs such that every 2-connected o-heavy and
-c-heavy graph is hamiltonian. Our work results in a different proof of a
stronger version of Hu's theorem. Furthermore, our main result improves or
extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones
Mathematicae Graph Theor
Hamilton cycles in almost distance-hereditary graphs
Let be a graph on vertices. A graph is almost
distance-hereditary if each connected induced subgraph of has the
property for any pair of vertices .
A graph is called 1-heavy (2-heavy) if at least one (two) of the end
vertices of each induced subgraph of isomorphic to (a claw) has
(have) degree at least , and called claw-heavy if each claw of has a
pair of end vertices with degree sum at least . Thus every 2-heavy graph is
claw-heavy. In this paper we prove the following two results: (1) Every
2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian.
(2) Every 3-connected, 1-heavy and almost distance-hereditary graph is
Hamiltonian. In particular, the first result improves a previous theorem of
Feng and Guo. Both results are sharp in some sense.Comment: 14 pages; 1 figure; a new theorem is adde
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