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80
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.
A short proof of the Hajnal–Szemerédi Theorem on equitable coloring
 Combin. Probab. Comput
, 2008
"... A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r +1 colours. The ..."
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Cited by 33 (9 self)
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A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r +1 colours. The proof yields a polynomial time algorithm for such colourings. 1.
GeneaQuilts: A System for Exploring Large Genealogies
 INRIA Nancy – Grand Est : LORIA, Technopôle de NancyBrabois  Campus scientifique 615, rue du Jardin Botanique  BP 101  54602 VillerslèsNancy Cedex Centre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau  Rocquencourt  BP 105  78153 L
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 25 (4 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
An Oretype theorem on equitable coloring
, 2008
"... A proper vertex coloring of a graph is equitable if the sizes of its color classes differ by at most one. In this paper, we prove that if G is a graph such that for each edge xy ∈ E(G), thesumd(x) + d(y) of the degrees of its ends is at most 2r + 1, then G has an equitable coloring with r + 1 colors ..."
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Cited by 15 (7 self)
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A proper vertex coloring of a graph is equitable if the sizes of its color classes differ by at most one. In this paper, we prove that if G is a graph such that for each edge xy ∈ E(G), thesumd(x) + d(y) of the degrees of its ends is at most 2r + 1, then G has an equitable coloring with r + 1 colors. This extends the Hajnal–Szemerédi Theorem on graphs with maximum degree r and a recent conjecture by Kostochka and Yu. We also pose an Oretype version of the Chen–Lih–Wu Conjecture and prove a very partial case of it.
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.
Spectral radius and Hamiltonicity of graphs
, 2009
"... Let G be a graph of order n and µ (G) be the largest eigenvalue of its adjacency matrix. Let G be the complement of G. Write Kn−1 + v for the complete graph on n − 1 vertices together with an isolated vertex, and Kn−1 + e for the complete graph on n − 1 vertices with a pendent edge. We show that: If ..."
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Cited by 6 (0 self)
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Let G be a graph of order n and µ (G) be the largest eigenvalue of its adjacency matrix. Let G be the complement of G. Write Kn−1 + v for the complete graph on n − 1 vertices together with an isolated vertex, and Kn−1 + e for the complete graph on n − 1 vertices with a pendent edge. We show that: If µ (G) ≥ n − 2, then G contains a Hamiltonian path unless G = Kn−1 + v; if strict inequality holds, then G contains a Hamiltonian cycle unless G = Kn−1 + e. If µ ( G) ≤ √ n − 1, then G contains a Hamiltonian path unless G = Kn−1 + v. If µ ( G) ≤ √ n − 2, then G contains a Hamiltonian cycle unless G = Kn−1 + e.
Language Reference
 In Microsoft Visual C++ 5.0 Programmer’s Reference Set, Volume 4. Of Microsoft Visual C++ Language Reference, Microsoft
, 1997
"... On the combinatorial characterization of quasicrystals. (English summary) ..."
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Cited by 5 (0 self)
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On the combinatorial characterization of quasicrystals. (English summary)
A DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS
"... We show that for each α> 0 every sufficiently large oriented graph G with δ + (G), δ − (G) ≥ 3G/8 + αG  contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G) + δ + (G) + δ − (G ..."
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Cited by 5 (4 self)
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We show that for each α> 0 every sufficiently large oriented graph G with δ + (G), δ − (G) ≥ 3G/8 + αG  contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G) + δ + (G) + δ − (G) ≥ 3G/2 + αG. Up to the term αG  this confirms a conjecture of Häggkvist [10]. We also prove an Oretype theorem for oriented graphs.