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Paths and cycles in colored graphs
 Australasian J. Combin
"... Let G be an (edge)colored graph. A path (cycle) is called monochromatic if all of its edges have the same color, and is called heterochromatic if all of its edges have different colors. In this paper, some sufficient conditions for the existence of (long) monochromatic paths and cycles, and those f ..."
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Cited by 23 (9 self)
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Let G be an (edge)colored graph. A path (cycle) is called monochromatic if all of its edges have the same color, and is called heterochromatic if all of its edges have different colors. In this paper, some sufficient conditions for the existence of (long) monochromatic paths and cycles, and those for the existences of long heterochromatic paths and cycles are obtained. It is proved that the problem of finding a path (cycle) with as few different colors as possible in a colored graph is NPhard. Several exact and approximation algorithms for finding a path with the fewest colors are provided. The complexity of the exact algorithms and the performance ratio of the approximation algorithms are analyzed. We also pose a problem on the existence of paths and cycles with many different colors.
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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Cited by 21 (1 self)
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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.
Hamiltonicity thresholds in Achlioptas processes
"... In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K = K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the cur ..."
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Cited by 18 (6 self)
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In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K = K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K = o(log n), the threshold for Hamiltonicity is 1+o(1) 2K n log n, i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K = ω(log n) we can essentially waste almost no edges, and create a Hamilton cycle in n + o(n) rounds with high probability. Finally, in the intermediate regime where K = Θ(log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3.
Almost spanning subgraphs of random graphs after adversarial edge removal
 Combin. Probab. Comput
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Long heterochromatic paths in edgecolored graphs
 Research Paper R33
"... Let G be an edgecolored graph. A heterochromatic path of G is such a path in which no two edges have the same color. dc (v) denotes the color degree of a vertex v of G. In a previous paper, we showed that if dc (v) ≥ k for every vertex v of G, thenGhasaheterochromatic path of length at least ⌈ k+1 ..."
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Let G be an edgecolored graph. A heterochromatic path of G is such a path in which no two edges have the same color. dc (v) denotes the color degree of a vertex v of G. In a previous paper, we showed that if dc (v) ≥ k for every vertex v of G, thenGhasaheterochromatic path of length at least ⌈ k+1 2 ⌉. It is easy to see that if k =1, 2, G has a heterochromatic path of length at least k. Saito conjectured that under the color degree condition G has a heterochromatic path of length at least ⌈ 2k+1 3 ⌉. Even if this is true, no one knows if it is a best possible lower bound. Although we cannot prove Saito’s conjecture, we can show in this paper that if 3 ≤ k ≤ 7, G has a heterochromatic path of length at least k − 1, and if k ≥ 8, G has a heterochromatic path of length at least ⌈ 3k 5 ⌉ + 1. Actually, we can show that for 1 ≤ k ≤ 5 any graph G under the color degree condition has a heterochromatic path of length at least k, with only one exceptional graph K4 for k = 3, one exceptional graph for k = 4 and three exceptional graphs for k =5,for which G has a heterochromatic path of length at least k−1. Our experience suggests us to conjecture that under the color degree condition G has a heterochromatic path of length at least k − 1. 1.
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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Cited by 9 (0 self)
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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 ⌈ ⌉
Connectivity threshold of Bluetooth graphs
, 2011
"... We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc ” wireless networks. The graphs are obtained as “irrigation subgraphs ” of the wellknown random geometric graph model. There are two parameters that control the model: the radius r that determines the “visibl ..."
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Cited by 6 (4 self)
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We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc ” wireless networks. The graphs are obtained as “irrigation subgraphs ” of the wellknown random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors ” of each node and the number of edges c that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph. 1
Rainbow Hamilton cycles in uniform hypergraphs
"... be the complete kuniform hypergraph, k ≥ 3, and let ℓ be an integer such that 1 ≤ ℓ ≤ k − 1 and k − ℓ divides n. An ℓoverlapping Hamilton cycle in K (k) n is a spanning subhypergraph C of K (k) n with n/(k − ℓ) edges and such that for some cyclic ordering of the vertices each edge of C consists of ..."
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be the complete kuniform hypergraph, k ≥ 3, and let ℓ be an integer such that 1 ≤ ℓ ≤ k − 1 and k − ℓ divides n. An ℓoverlapping Hamilton cycle in K (k) n is a spanning subhypergraph C of K (k) n with n/(k − ℓ) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of adjacent edges in C intersects in precisely ℓ vertices. We show that, for some constant c = c(k, ℓ) and sufficiently large n, for every coloring (partition) of the edges of K (k) n which uses arbitrarily many colors but no color appears more than cn k−ℓ times, there exists a rainbow ℓoverlapping Hamilton cycle C, that is every edge of C receives a different color. We also prove that, for some constant c ′ = c ′ (k, ℓ) and sufficiently large n, for every coloring of the edges of K (k) n in which the maximum degree of the subhypergraph induced by any single color is bounded by c ′ n k−ℓ, there exists a properly colored ℓoverlapping Hamilton cycle C, that is every two adjacent edges receive different colors. For ℓ = 1, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for 2 ≤ ℓ ≤ k − 1. The proofs rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.
Multicoloured Hamilton cycles in random graphs; an antiRamsey threshold
, 1995
"... Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a kbounded colouring. We say that a subset of the edges of G is multicoloured if each edge is of a different colour. We say that the colouring is Hgood, if a multicoloured Hamilton cycle exists ..."
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Cited by 6 (3 self)
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Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a kbounded colouring. We say that a subset of the edges of G is multicoloured if each edge is of a different colour. We say that the colouring is Hgood, if a multicoloured Hamilton cycle exists i.e., one with a multicoloured edgeset. Let AR k = fG : every kbounded colouring of G is Hgoodg. We establish the threshold for the random graph G n;m to be in AR k .