1,092 research outputs found
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
Assessing the Computational Complexity of Multi-Layer Subgraph Detection
Multi-layer graphs consist of several graphs (layers) over the same vertex
set. They are motivated by real-world problems where entities (vertices) are
associated via multiple types of relationships (edges in different layers). We
chart the border of computational (in)tractability for the class of subgraph
detection problems on multi-layer graphs, including fundamental problems such
as maximum matching, finding certain clique relaxations (motivated by community
detection), or path problems. Mostly encountering hardness results, sometimes
even for two or three layers, we can also spot some islands of tractability
Hamiltonian chordal graphs are not cycle extendible
In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle
extendible; that is, the vertices of any non-Hamiltonian cycle are contained in
a cycle of length one greater. We disprove this conjecture by constructing
counterexamples on vertices for any . Furthermore, we show that
there exist counterexamples where the ratio of the length of a non-extendible
cycle to the total number of vertices can be made arbitrarily small. We then
consider cycle extendibility in Hamiltonian chordal graphs where certain
induced subgraphs are forbidden, notably and the bull.Comment: Some results from Section 3 were incorrect and have been removed. To
appear in SIAM Journal on Discrete Mathematic
Forbidden subgraphs that imply Hamiltonian-connectedness
It is proven that if is a -connected claw-free graph which is also -free (where is a triangle with a path of length attached), -free (where is a path with vertices) or -free (where consists of two disjoint triangles connected by an edge), then is Hamiltonian-connected. Also, examples will be described that determine a finite family of graphs such that if a 3-connected graph being claw-free and -free implies is Hamiltonian-connected, then . \u
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