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153,133
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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vertex, and for indifference graphs with odd maximum degree. For the latter subclass, we prove that Ø 0 = \Delta. 1 Introduction An edgecolouring of a graph is an assignment of colours to its edges such that no adjacent edges have the same colour. The chromatic index of a Universidade Federal do
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
Kempe equivalence of edgecolourings in subcubic and subquartic graphs
, 2010
"... It is proved that all 4edgecolourings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree ∆ = 3 is a threshold for Kempe equivalence of (∆+1)edgecolourings, as such an equivalence does not hold in general when ..."
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It is proved that all 4edgecolourings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree ∆ = 3 is a threshold for Kempe equivalence of (∆+1)edgecolourings, as such an equivalence does not hold in general
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 Critical Point For Random Graphs With A Given Degree Sequence
, 2000
"... Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 the ..."
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Cited by 511 (8 self)
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Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0
The 3edgecolouring problem on the
, 2009
"... We consider the problem of counting the number of 3colourings of the edges (bonds) of the 48 lattice and the 312 lattice. These lattices are Archimedean with coordination number 3, and can be regarded as decorated versions of the square and honeycomb lattice, respectively. We solve these edgecol ..."
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We consider the problem of counting the number of 3colourings of the edges (bonds) of the 48 lattice and the 312 lattice. These lattices are Archimedean with coordination number 3, and can be regarded as decorated versions of the square and honeycomb lattice, respectively. We solve these edgecolouring
"GrabCut”  interactive foreground extraction using iterated graph cuts
 ACM TRANS. GRAPH
, 2004
"... The problem of efficient, interactive foreground/background segmentation in still images is of great practical importance in image editing. Classical image segmentation tools use either texture (colour) information, e.g. Magic Wand, or edge (contrast) information, e.g. Intelligent Scissors. Recently ..."
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Cited by 1140 (36 self)
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The problem of efficient, interactive foreground/background segmentation in still images is of great practical importance in image editing. Classical image segmentation tools use either texture (colour) information, e.g. Magic Wand, or edge (contrast) information, e.g. Intelligent Scissors
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
Multicoloured Hamilton cycles in random edgecoloured graphs
, 2002
"... We define a space of random edgecoloured graphs n,m,n which correspond naturally to edge ncolourings of Gn,m. We show that there exist constants K0, K1 _ 21 such that provided m _ Kon log n and n _ Kin then a random edge coloured graph contains a multicoloured Hamilton cycle with probability ..."
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Cited by 6 (3 self)
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We define a space of random edgecoloured graphs n,m,n which correspond naturally to edge ncolourings of Gn,m. We show that there exist constants K0, K1 _ 21 such that provided m _ Kon log n and n _ Kin then a random edge coloured graph contains a multicoloured Hamilton cycle with probability
Results 1  10
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153,133