174 research outputs found

    Forbidden subgraphs that imply Hamiltonian-connectedness

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    It is proven that if GG is a 33-connected claw-free graph which is also Z3Z_3-free (where Z3Z_3 is a triangle with a path of length 33 attached), P6P_6-free (where P6P_6 is a path with 66 vertices) or H1H_1-free (where H1H_1 consists of two disjoint triangles connected by an edge), then GG is Hamiltonian-connected. Also, examples will be described that determine a finite family of graphs L\cal{L} such that if a 3-connected graph being claw-free and LL-free implies GG is Hamiltonian-connected, then LLL\in\cal{L}. \u

    Forbidden subgraphs that imply hamiltonian-connectedness

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    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 L{\cal L}equation image such that if a 3‐connected graph being claw‐free and L‐free implies G is hamiltonian‐connected, then L L\in \cal Lequation imag

    Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2-connected graphs

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    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 GG be a graph on nn vertices and HH be an induced subgraph of GG. HH is called \emph{o}-heavy if there are two nonadjacent vertices in HH with degree sum at least nn, and is called ff-heavy if for every two vertices u,vV(H)u,v\in V(H), dH(u,v)=2d_{H}(u,v)=2 implies that max{d(u),d(v)}n/2\max\{d(u),d(v)\}\geq n/2. We say that GG is HH-\emph{o}-heavy (HH-\emph{f}-heavy) if every induced subgraph of GG isomorphic to HH is \emph{o}-heavy (\emph{f}-heavy). In this paper we characterize all connected graphs RR and SS other than P3P_3 such that every 2-connected RR-\emph{f}-heavy and SS-\emph{f}-heavy (RR-\emph{o}-heavy and SS-\emph{f}-heavy, RR-\emph{f}-heavy and SS-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

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    Let GG be a graph. Adopting the terminology of Broersma et al. and \v{C}ada, respectively, we say that GG is 2-heavy if every induced claw (K1,3K_{1,3}) of GG contains two end-vertices each one has degree at least V(G)/2|V(G)|/2; and GG is o-heavy if every induced claw of GG contains two end-vertices with degree sum at least V(G)|V(G)| in GG. In this paper, we introduce a new concept, and say that GG is \emph{SS-c-heavy} if for a given graph SS and every induced subgraph GG' of GG isomorphic to SS and every maximal clique CC of GG', every non-trivial component of GCG'-C contains a vertex of degree at least V(G)/2|V(G)|/2 in GG. 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 NN-c-heavy graph is hamiltonian, where NN is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs SS such that every 2-connected o-heavy and SS-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

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    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

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    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

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    We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, tt-tough graphs, and claw-free graphs

    Global cycle properties in graphs with large minimum clustering coefficient

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    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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