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33
Monochromatic and heterochromatic subgraphs in edgecolored graphsa survey
 Graphs and Combinatorics
, 2008
"... Abstract. Let Kn be the complete graph with n vertices and c1,c2, · · ·,cr be r different colors. Suppose we randomly and uniformly color the edges of Kn in c1,c2, · · ·,cr. Then we get a random graph, denoted by Kr n. In the paper, we investigate the asymptotic properties of several kinds of ..."
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Abstract. Let Kn be the complete graph with n vertices and c1,c2, · · ·,cr be r different colors. Suppose we randomly and uniformly color the edges of Kn in c1,c2, · · ·,cr. Then we get a random graph, denoted by Kr n. In the paper, we investigate the asymptotic properties of several kinds of monochromatic and heterochromatic subgraphs in Kr n. Accurate threshold functions in some cases are also obtained.
Constrained Ramsey numbers of graphs
 J. GRAPH THEORY
, 2003
"... ... In this paper we determine R(F) for all families consisting of equipartitioned stars, and we prove that 5b s\Gamma 12 c+1 ^ R(F) ^ 3s\Gamma p3s when F consists of a monochromatic star of size s and a polychromatic triangle. ..."
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... In this paper we determine R(F) for all families consisting of equipartitioned stars, and we prove that 5b s\Gamma 12 c+1 ^ R(F) ^ 3s\Gamma p3s when F consists of a monochromatic star of size s and a polychromatic triangle.
Heterochromatic matchings in edgecolored graphs
 2008), Paper #R138. the electronic journal of combinatorics 17 (2010), #N26 5
"... Let G be an (edge)colored graph. A heterochromatic matching of G is a matching in which no two edges have the same color. For a vertex v, let dc (v) be the color degree of v. We show that if dc (v) ≥ k for every vertex v of G, then G has a heterochromatic matching of size ⌈ ⌉ ..."
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Let G be an (edge)colored graph. A heterochromatic matching of G is a matching in which no two edges have the same color. For a vertex v, let dc (v) be the color degree of v. We show that if dc (v) ≥ k for every vertex v of G, then G has a heterochromatic matching of size ⌈ ⌉
Sufficient conditions for the existence of perfect heterochromatic matchings in colored graphs
, 2008
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Rainbow Arithmetic Progressions and AntiRamsey Results
, 2002
"... The van der Waerden theorem in Ramsey theory states that for every k and t and suciently large N , every kcoloring of [N ] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoicic conjectured that every equinumerous 3coloring of [3n] contains a 3term ra ..."
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Cited by 8 (3 self)
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The van der Waerden theorem in Ramsey theory states that for every k and t and suciently large N , every kcoloring of [N ] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoicic conjectured that every equinumerous 3coloring of [3n] contains a 3term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are colored with distinct colors. In this paper, we prove that every 3coloring of the set of natural numbers for which each color class has density more than 1=6, contains a 3term rainbow arithmetic progression. We also prove similar results for colorings of Zn . Finally, we give a general perspective on other antiRamseytype problems that can be considered.
RAMSEYTYPE PROBLEM FOR AN ALMOST MONOCHROMATIC K_4
, 2008
"... In this short note we prove that there is a constant c such that every kedgecoloring of the complete graph K_n with n ≥ 2^ck contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem. ..."
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In this short note we prove that there is a constant c such that every kedgecoloring of the complete graph K_n with n ≥ 2^ck contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.
Rainbow Ramsey Theory
, 2004
"... This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic progressions in [n] and N. A general perspective on other rainbow Ramsey type problems is given. ..."
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Cited by 5 (1 self)
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This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic progressions in [n] and N. A general perspective on other rainbow Ramsey type problems is given.
PROPERLY COLOURED COPIES AND RAINBOW COPIES OF LARGE GRAPHS WITH SMALL MAXIMUM DEGREE
, 2010
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Rainbow Turán problems
 Combin. Probab. Comput
"... For a fixed graph H, we define the rainbow Turán number ex ∗ (n, H) to be the maximum number of edges in a graph on n vertices that has a proper edgecolouring with no rainbow H. Recall that the (ordinary) Turán number ex(n, H) is the maximum number of edges in a graph on n vertices that does not co ..."
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For a fixed graph H, we define the rainbow Turán number ex ∗ (n, H) to be the maximum number of edges in a graph on n vertices that has a proper edgecolouring with no rainbow H. Recall that the (ordinary) Turán number ex(n, H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any nonbipartite H we show that ex ∗ (n, H) = (1+o(1))ex(n, H), and if H is colourcritical we show that ex ∗ (n, H) = ex(n, H). When H is the complete bipartite graph Ks,t with s ≤ t we show ex ∗ (n, Ks,t) = O(n 2−1/s), which matches the known bounds for ex(n, Ks,t) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex ∗ (n, C6) = O(n 4/3), which is of the correct order of magnitude. 1
The heterochromatic matchings in edgecolored bipartite graphs
"... Let (G, C) be an edgecolored bipartite graph with bipartition (X, Y). A heterochromatic matching of G is such a matching in which no two edges have the same color. Let N c (S) denote a maximum color neighborhood of S ⊆ V (G). We show that if N c (S)  ≥ S  for all S ⊆ X, then G has a heterochro ..."
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Let (G, C) be an edgecolored bipartite graph with bipartition (X, Y). A heterochromatic matching of G is such a matching in which no two edges have the same color. Let N c (S) denote a maximum color neighborhood of S ⊆ V (G). We show that if N c (S)  ≥ S  for all S ⊆ X, then G has a heterochromatic matching with cardinality at least ⌈ X 3 ⌉. We also obtain that if X  = Y  = n and N c (S)  ≥ S for all S ⊆ X or S ⊆ Y, then G has a heterochromatic matching with cardinality at least ⌈ 3n−1 8 ⌉.