1,777 research outputs found

    Hamilton cycles in almost distance-hereditary graphs

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    Let GG be a graph on n≥3n\geq 3 vertices. A graph GG is almost distance-hereditary if each connected induced subgraph HH of GG has the property dH(x,y)≤dG(x,y)+1d_{H}(x,y)\leq d_{G}(x,y)+1 for any pair of vertices x,y∈V(H)x,y\in V(H). A graph GG is called 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of GG isomorphic to K1,3K_{1,3} (a claw) has (have) degree at least n/2n/2, and called claw-heavy if each claw of GG has a pair of end vertices with degree sum at least nn. 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

    Some local--global phenomena in locally finite graphs

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    In this paper we present some results for a connected infinite graph GG with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of GG. (For a vertex ww of a graph GG the ball of radius rr centered at ww is the subgraph of GG induced by the set Mr(w)M_r(w) of vertices whose distance from ww does not exceed rr). In particular, we prove that if every ball of radius 2 in GG is 2-connected and GG satisfies the condition dG(u)+dG(v)≥∣M2(w)∣−1d_G(u)+d_G(v)\geq |M_2(w)|-1 for each path uwvuwv in GG, where uu and vv are non-adjacent vertices, then GG has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in GG satisfies Ore's condition (1960) then all balls of any radius in GG are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

    On factors of 4-connected claw-free graphs

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    We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is Hamiltonian, i.e. has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e. graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjecture 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths. \u

    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
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