1,776 research outputs found

    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 G′G' of GG isomorphic to SS and every maximal clique CC of G′G', every non-trivial component of G′−CG'-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

    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

    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

    Degree and neighborhood conditions for hamiltonicity of claw-free graphs

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    For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers p ≥ t \u3e 0 , k ∈ { 2 , 3 } and N = N ( p , ϵ ) , if H is a k -connected claw-free graph of order n \u3e N with δ ( H ) ≥ 3 and its Ryjác̆ek’s closure c l ( H ) = L ( G ) , and if d t ( H ) ≥ t ( n + ϵ ) ∕ p where d t ( H ) ∈ { σ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p − 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: (i) If k = 2 , δ ( H ) ≥ 3 , and for a given t ( 1 ≤ t ≤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ≥ 1 ). When t = 4 and d t ( H ) = σ 4 ( H ) , this proves a conjecture in Frydrych (2001). (ii) If k = 3 , δ ( H ) ≥ 24 , and for a given t ( 1 ≤ t ≤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p − 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer
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