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On EdgeColouring Indifference Graphs
, 1994
"... Vizing's theorem states that the chromatic index Ø 0 (G) of a graph G is either the maximum degree \Delta(G) or \Delta(G) + 1. A graph G is called overfull if jE(G)j ? \Delta(G)bjV (G)j=2c. A sufficient condition for Ø 0 (G) = \Delta(G) +1 is that G contains an overfull subgraph H with \Del ..."
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Vizing's theorem states that the chromatic index Ø 0 (G) of a graph G is either the maximum degree \Delta(G) or \Delta(G) + 1. A graph G is called overfull if jE(G)j ? \Delta(G)bjV (G)j=2c. A sufficient condition for Ø 0 (G) = \Delta(G) +1 is that G contains an overfull subgraph H
Matching, EdgeColouring, Dimers
"... Abstract. We survey some recent results on finding and counting perfect matchings in regular bipartite graphs, with applications to bipartite edgecolouring and the dimer constant. Main results are improved complexity bounds for finding a perfect matching in a regular bipartite graph and for edgeco ..."
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Abstract. We survey some recent results on finding and counting perfect matchings in regular bipartite graphs, with applications to bipartite edgecolouring and the dimer constant. Main results are improved complexity bounds for finding a perfect matching in a regular bipartite graph and for edgecolouring
On EdgeColouring Indifference Graphs
, 1997
"... . Vizing's theorem states that the chromatic index Ø 0 (G) of a graph G is either the maximum degree \Delta(G) or \Delta(G) + 1. A graph G is called overfull if jE(G)j ? \Delta(G)bjV (G)j=2c. A sufficient condition for Ø 0 (G) = \Delta(G) + 1 is that G contains an overfull subgraph H wit ..."
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Cited by 3 (3 self)
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. Vizing's theorem states that the chromatic index Ø 0 (G) of a graph G is either the maximum degree \Delta(G) or \Delta(G) + 1. A graph G is called overfull if jE(G)j ? \Delta(G)bjV (G)j=2c. A sufficient condition for Ø 0 (G) = \Delta(G) + 1 is that G contains an overfull subgraph H
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices
Books in graphs
, 2008
"... A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) ..."
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Cited by 2380 (22 self)
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A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α
Large rainbow matchings in edgecoloured graphs
 Combinatorics, Probability, and Computing
"... A rainbow subgraph of an edgecoloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k � 4, every edgecoloured graph with minimum colour degree k contains a rainbow ..."
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Cited by 6 (1 self)
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rainbow matching of size at least ⌈k/2⌉. A properly edgecoloured K4 has no such matching, which motivates the restriction k � 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2
Coloured matchings in edgecoloured graphs
, 2007
"... Erdős � and Gallai proved that every graph of order n with more than f(k,n) = �2k−1� �k−1 � � max 2, 2 + (k − 1)(n − k + 1) edges contains a matching with k edges. We generalize this and show that if R (red) and B (blue) are graphs on the same vertex set of size n, each with more than f(k,n) edges, ..."
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, then the edgecoloured multigraph R ∪ B contains any 2edgecoloured matching with k edges. In general, we prove that for n ≥ 3k − 1, if G1,G2,...,Gt are graphs on the same vertex set of size n, such that the edges of Gi are coloured i and Gi has more than f(k,n) edges for all 1 ≤ i ≤ t, then the edgecoloured
The NPCompleteness of EdgeColouring
, 1981
"... We show that it is NPcomplete to determine the chromatic index of an arbitrary graph. The problem remains NPcomplete even for cubic graphs. ..."
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Cited by 32 (0 self)
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We show that it is NPcomplete to determine the chromatic index of an arbitrary graph. The problem remains NPcomplete even for cubic graphs.
Gadget Graphs for Edgecoloured Graphs
, 805
"... In this paper, we consider properly edgecoloured (PC) paths and cycles in edgecoloured graphs. We consider a family of transformations of an edgecoloured graph G into an ordinary graph that allow us to check the existence PC cycles and PC (s, t)paths in G and, if they exist, to find shortest one ..."
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In this paper, we consider properly edgecoloured (PC) paths and cycles in edgecoloured graphs. We consider a family of transformations of an edgecoloured graph G into an ordinary graph that allow us to check the existence PC cycles and PC (s, t)paths in G and, if they exist, to find shortest
Results 1  10
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