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305,070
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
Homomorphisms and Edgecolourings of Planar Graphs
"... We conjecture that every planar graph of oddgirth 2k + 1 admits a homomorphism to Cayley graph C(Z 2k+1 2, S2k+1), with S2k+1 being the set of (2k + 1)vectors with exactly two consecutive 1’s in a cyclic order. This is an strengthening of a conjecture of T. Marshall, J. Neˇsetˇril and the author. O ..."
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Cited by 2 (1 self)
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. Our main result is to show that this conjecture is equivalent to the corresponding case of a conjecture of P. Seymour, stating that every planar (2k +1)graph is (2k +1)edgecolourable. 1
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
A Note on Alternating Cycles in Edgecoloured Graphs
 J.Combin. Theory Ser B
, 1996
"... Grossman and Haggkvist gave a characterisation of twoedgecoloured graphs, which have an alternating cycle (i.e. a cycle in which no two consecutive edges have the same colour). We extend their characterisation to edgecoloured graphs, with any number of colours. That is we show that if there is no ..."
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Cited by 10 (0 self)
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that if there is no alternating cycle in an edgecoloured graph, G, then it contains a vertex z, such that there is no connected component of G \Gamma z, which is joined to z with edges of different colors. Our extension implies a polynomial algorithm, for deciding if an edgecoloured graph contains an alternating cycle (see
On parcimonious edgecolouring of graphs with maximum degree three
, 2012
"... In a graph G of maximum degree ∆ let γ denote the largest fraction of edges that can be ∆ edgecoloured. Albertson and Haas showed that γ ≥ 13 when G is cubic. We show here that this result can be extended 15 to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, ther ..."
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Cited by 6 (5 self)
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In a graph G of maximum degree ∆ let γ denote the largest fraction of edges that can be ∆ edgecoloured. Albertson and Haas showed that γ ≥ 13 when G is cubic. We show here that this result can be extended 15 to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover
A code for mbipartite edgecoloured graphs
 Rend. Istit. Mat. Univ. Trieste
, 2002
"... Summary. An (n + 1)coloured graph (Γ, γ) is said to be mbipartite if m is the maximum integer so that every mresidue of (Γ, γ) (i.e. every connected subgraph whose edges are coloured by only m colours) is bipartite; obviously, every (n+ 1)coloured graph, with n ≥ 2, results to be mbipartite fo ..."
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Cited by 4 (0 self)
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transforms the graphs one into the other, up to permutation of the edgecolouring. More precisely, if H is a given group of permutations on the colour set, we face the problem of algorithmically recognizing Hisomorphic coloured graphs by means of a suitable definition of Hcode.
Homomorphisms of Edgecoloured Graphs and Coxeter Groups
"... Let G1 = (V1, E1) and G2 = (V2, E2) be two edgecoloured graphs (without multiple edges or loops). A homomorphism is a mapping φ: V1 ↦− → V2 for which, for every pair of adjacent vertices u and v of G1, φ(u) and φ(v) are adjacent in G2 and the colour of the edge φ(u)φ(v) is the same as that of the e ..."
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Let G1 = (V1, E1) and G2 = (V2, E2) be two edgecoloured graphs (without multiple edges or loops). A homomorphism is a mapping φ: V1 ↦− → V2 for which, for every pair of adjacent vertices u and v of G1, φ(u) and φ(v) are adjacent in G2 and the colour of the edge φ(u)φ(v) is the same
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
Results 1  10
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305,070