Results 1  10
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33
Embedding large subgraphs into dense graphs
, 2009
"... What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings. Perfect matchings are generalized by perfect Fpackings, where instead of covering ..."
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Cited by 34 (11 self)
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What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings. Perfect matchings are generalized by perfect Fpackings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect Fpacking, so as in the case of Dirac’s theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect Fpacking. The Regularity lemma of Szemerédi and the Blowup lemma of Komlós, Sárközy and Szemerédi have proved to be powerful tools in attacking such problems and quite recently, several longstanding problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on Fpackings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
On Sufficient Degree Conditions for a Graph to be klinked
, 2005
"... A graph is klinked if for every list of 2k vertices {s1,...,sk,t1,...,tk}, there exist internally disjoint paths P1,...,Pk such that each Pi is an si,tipath. We consider degree conditions and connectivity conditions sufficient to force a graph to be klinked. Let D(n, k) be the minimum positive in ..."
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Cited by 13 (4 self)
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A graph is klinked if for every list of 2k vertices {s1,...,sk,t1,...,tk}, there exist internally disjoint paths P1,...,Pk such that each Pi is an si,tipath. We consider degree conditions and connectivity conditions sufficient to force a graph to be klinked. Let D(n, k) be the minimum positive integer d such that every nvertex graph with minimum degree at least d is klinked and let R(n, k) be the minimum positive integer r such that every nvertex graph in which the sum of degrees of each pair of nonadjacent vertices is at least r is klinked. The main result of the paper is finding the exact values of D(n, k) andR(n, k) for every n and k. Thomas and Wollan [14] used the bound D(n, k) � (n +3k)/2 − 2 to give sufficient conditions for a graph to be klinked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every 2kconnected graph with average degree at least 12k is klinked.
A SURVEY ON HAMILTON CYCLES IN DIRECTED GRAPHS
"... We survey some recent results on longstanding conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly’s conjecture on Hamilton decompositions of regular tournaments: the edges of ..."
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Cited by 10 (8 self)
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We survey some recent results on longstanding conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly’s conjecture on Hamilton decompositions of regular tournaments: the edges of every regular tournament can be covered by a set of Hamilton cycles which are ‘almost’ edgedisjoint. We also highlight the role that the notion of ‘robust expansion’ plays in several of the proofs. New and old open problems are discussed.
On packing Hamilton Cycles in ɛRegular Graphs
, 2003
"... A graph G = (V; E) on n vertices is (; )regular if its minimal degree is at least n, and for every pair of disjoint subsets S; T V of cardinalities at least n, the number of edges e(S; T ) between S and T satis es: e(S;T ) . We prove that if > 0 are constants, then every (; )regul ..."
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Cited by 8 (7 self)
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A graph G = (V; E) on n vertices is (; )regular if its minimal degree is at least n, and for every pair of disjoint subsets S; T V of cardinalities at least n, the number of edges e(S; T ) between S and T satis es: e(S;T ) . We prove that if > 0 are constants, then every (; )regular graph on n vertices contains a family of (=2 O())n edgedisjoint Hamilton cycles. As a consequence we derive that for every constant 0 < p < 1, with high probability in the random graph G(n; p), almost all edges can be packed into edgedisjoint Hamilton cycles. A similar result is proven for the directed case.
Not being (super)thin or solid is hard: A study of grid Hamiltonicity
, 2008
"... We give a systematic study of Hamiltonicity of grids—the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, “square”, grids, we give a compr ..."
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Cited by 3 (0 self)
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We give a systematic study of Hamiltonicity of grids—the graphs induced by finite subsets of vertices of the tilings of the plane with congruent regular convex polygons (triangles, squares, or hexagons). Summarizing and extending existing classification of the usual, “square”, grids, we give a comprehensive taxonomy of the grid graphs. For many classes of grid graphs we resolve the computational complexity of the Hamiltonian cycle problem. For graphs for which there exists a polynomialtime algorithm we give efficient algorithms to find a Hamiltonian cycle. We also establish, for any g ≥ 6, a onetoone correspondence between Hamiltonian cycles in planar bipartite maximumdegree3 graphs and Hamiltonian cycles in the class Cg of girthg planar maximumdegree3 graphs. As applications of the correspondence, we show that for graphs in Cg the Hamiltonian cycle problem is NPcomplete and that for any N ≥ 5 there exist graphs in Cg that have exactly N Hamiltonian cycles. We also prove that for the graphs in Cg, a Chinese Postman tour gives a (1 + 8 g)approximation to TSP, improving thereby the Christofides ratio when g> 16. We show further that, on any graph, the tour obtained by Christofides ’ algorithm is not longer than a Chinese Postman tour.
EMBEDDING CYCLES IN FINITE PLANES
, 1305
"... Abstract. We define and study embeddings of cycles in finite affine and projective planes. We show that for all k, 3 ≤ k ≤ q 2, a kcycle can be embedded in any affine plane of order q. We also prove a similar result for finite projective planes: for all k, 3 ≤ k ≤ q 2 + q + 1, a kcycle can be embe ..."
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Cited by 3 (2 self)
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Abstract. We define and study embeddings of cycles in finite affine and projective planes. We show that for all k, 3 ≤ k ≤ q 2, a kcycle can be embedded in any affine plane of order q. We also prove a similar result for finite projective planes: for all k, 3 ≤ k ≤ q 2 + q + 1, a kcycle can be embedded in any projective plane of order q. 1.
Necessary and sufficient conditions for unit graphs to be Hamiltonian
, 2011
"... The unit graph corresponding to an associative ring R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit of R. By a constructive method, we derive necessary and sufficient conditions for unit ..."
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Cited by 3 (3 self)
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The unit graph corresponding to an associative ring R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit of R. By a constructive method, we derive necessary and sufficient conditions for unit graphs to be Hamiltonian.
Closure for the Property of Having a Hamiltonian Prism
, 2003
"... We prove that a graph G of order n has a hamiltonian prism if and only if the graph Cl 4n=3 4=3 (G) has a hamiltonian prism where Cl 4n=3 4=3 (G) is the graph obtained from G by sequential adding edges between nonadjacent vertices whose degree sum is at least 4n=3 4=3. We show that this cannot ..."
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Cited by 2 (1 self)
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We prove that a graph G of order n has a hamiltonian prism if and only if the graph Cl 4n=3 4=3 (G) has a hamiltonian prism where Cl 4n=3 4=3 (G) is the graph obtained from G by sequential adding edges between nonadjacent vertices whose degree sum is at least 4n=3 4=3. We show that this cannot be improved to more than 4n=3 5.
Pancyclicity of Hamiltonian and highly connected graphs
"... A celebrated theorem of Chvátal and Erdős says that G is Hamiltonian if κ(G) ≥ α(G), where κ(G) denotes the vertex connectivity and α(G) the independence number of G. Moreover, Bondy suggested that almost any nontrivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motiva ..."
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A celebrated theorem of Chvátal and Erdős says that G is Hamiltonian if κ(G) ≥ α(G), where κ(G) denotes the vertex connectivity and α(G) the independence number of G. Moreover, Bondy suggested that almost any nontrivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if κ(G) ≥ 600α(G) then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree δ(G) ≥ 600α(G) then G is pancyclic. Improving an old result of Erdős, we also show that G is pancyclic if it is Hamiltonian and n ≥ 150α(G) 3. Our arguments use the following theorem of independent interest on cycle lengths in graphs: if δ(G) ≥ 300α(G) then G contains a cycle of length ℓ for all 3 ≤ ℓ ≤ δ(G)/81. 1