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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 21 (8 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.
Embedding large subgraphs into dense graphs
"... Abstract. 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 o ..."
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Cited by 19 (12 self)
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Abstract. 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 11 (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.
New conditions for kordered Hamiltonian graphs
 ARS COMBINATORIA
, 2003
"... For a positive integer k, a graph G is kordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 k n / 2, and deg(u) þ deg(v) n þ (3k 9)/2 for ever ..."
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Cited by 10 (2 self)
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For a positive integer k, a graph G is kordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 k n / 2, and deg(u) þ deg(v) n þ (3k 9)/2 for every pair u; v of nonadjacent vertices of G, then G is kordered hamiltonian. Minimum degree conditions are also given for
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 7 (5 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.
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)
An Oretype analogue of the SauerSpencer Theorem
, 2007
"... Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if �(G1)�(G2) <0.5n, then G1 and G2 pack. In this note, we study an Oretype analogue of the Sauer–Spencer Theor ..."
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Cited by 4 (4 self)
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Two graphs G1 and G2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if �(G1)�(G2) <0.5n, then G1 and G2 pack. In this note, we study an Oretype analogue of the Sauer–Spencer Theorem. Let θ(G) = max{d(u) + d(v) : uv ∈ E(G)}. We show that if θ(G1)�(G2) <n, then G1 and G2 pack. We also characterize the pairs (G1,G2) of nvertex graphs satisfying θ(G1)�(G2) = n that do not pack.
A DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS
, 709
"... Abstract. 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 [16]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G)+δ + (G)+δ − ( ..."
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Cited by 3 (3 self)
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Abstract. 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 [16]. 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]. 1.
A note on Arbitrarily vertex decomposable graphs, Opuscula Mathematica 26
, 2006
"... A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1,...,nk) of positive integers such that n1 +... + nk = n there exists a partition (V1,...,Vk) of the vertex set of G such that for each i ∈ {1,...,k}, Vi induces a connected subgraph of G on ni vertices. In th ..."
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Cited by 3 (0 self)
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A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1,...,nk) of positive integers such that n1 +... + nk = n there exists a partition (V1,...,Vk) of the vertex set of G such that for each i ∈ {1,...,k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a twoconnected graph on n vertices with the independence number at most ⌈n/2 ⌉ and such that the degree sum of any pair of nonadjacent vertices is at least n − 3, then G is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition where the bound n − 3 is replaced by n − 2. 1