174 research outputs found
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
Forbidden subgraphs that imply hamiltonian-connectedness
It is proven that if G is a 3‐connected claw‐free graph which is also H1‐free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian‐connected. Also, examples will be described that determine a finite family of graphs equation image such that if a 3‐connected graph being claw‐free and L‐free implies G is hamiltonian‐connected, then L equation imag
Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs
Bedrossian characterized all pairs of forbidden subgraphs for a 2-connected
graph to be Hamiltonian. Instead of forbidding some induced subgraphs, we relax
the conditions for graphs to be Hamiltonian by restricting Ore- and Fan-type
degree conditions on these induced subgraphs. Let be a graph on
vertices and be an induced subgraph of . is called \emph{o}-heavy if
there are two nonadjacent vertices in with degree sum at least , and is
called -heavy if for every two vertices ,
implies that . We say that is -\emph{o}-heavy
(-\emph{f}-heavy) if every induced subgraph of isomorphic to is
\emph{o}-heavy (\emph{f}-heavy). In this paper we characterize all connected
graphs and other than such that every 2-connected
-\emph{f}-heavy and -\emph{f}-heavy (-\emph{o}-heavy and
-\emph{f}-heavy, -\emph{f}-heavy and -free) graph is Hamiltonian. Our
results extend several previous theorems on forbidden subgraph conditions and
heavy subgraph conditions for Hamiltonicity of 2-connected graphs.Comment: 21 pages, 2 figure
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
Pairs of forbidden induced subgraphs for homogeneously traceable graphs
A graph G is called homogeneously traceable if for every vertex v of G, G contains a Hamilton path starting from v. For a graph H, we say that G is H-free if G contains no induced subgraph isomorphic to H. For a family H of graphs, G is called H-free if G is H-free for every H∈H. Determining families of graphs H such that every H-free graph G has some graph property has been a popular research topic for several decades, especially for Hamiltonian properties, and more recently for properties related to the existence of graph factors. In this paper we give a complete characterization of all pairs of connected graphs R,S such that every 2-connected {R,S}-free graph is homogeneously traceable
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
Global cycle properties in graphs with large minimum clustering coefficient
The clustering coefficient of a vertex in a graph is the proportion of
neighbours of the vertex that are adjacent. The minimum clustering coefficient
of a graph is the smallest clustering coefficient taken over all vertices. A
complete structural characterization of those locally connected graphs, with
minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully
cycle extendable is given in terms of strongly induced subgraphs with given
attachment sets. Moreover, it is shown that all locally connected graphs with
minimum clustering coefficient 1/2 and maximum degree at most 6 are weakly
pancyclic, thereby proving Ryjacek's conjecture for this class of locally
connected graphs.Comment: 16 pages, two figure
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