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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 with \De ..."
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Cited by 4 (4 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 with \Delta(H ) = \Delta(G). Plantholt proved that this condition is necessary for graphs with a universal vertex. In this paper, we conjecture that, for indifference graphs, this is also true. As supporting evidence, we prove this conjecture for general graphs with three maximal cliques and with no universal 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 graph is the minimum number of colours required to produce an edgecolouring for that graph. In this paper, we address...
Chromatic Index Critical Graphs of Orders 11 and 12
, 1997
"... A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results thi ..."
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Cited by 3 (1 self)
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A chromaticindexcritical graph G on n vertices is nontrivial if it has at most \Deltab n 2 c edges. We prove that there is no chromaticindexcritical graph of order 12, and that there are precisely two nontrivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three nontrivial chromaticindex critical graphs of order 12. 1 Introduction A famous theorem of Vizing [20] states that the chromatic index Ø 0 (G) of a simple graph G is \Delta(G) or \Delta(G) + 1, where \Delta(G) denotes the maximum vertex degree in G. A graph G is class 1 if Ø 0 (G) = \Delta(G) and it is class 2 otherwise. A class 2 graph G is (chromatic index) critical if Ø 0 (G \Gamma e) ! Ø 0 (G) for each edge e of G. If we want to stress the maximum vertex degree of a critical graph G we say G is \Delta(G)critical. Critical graphs of odd order are easy to construct while not much is known about critical graphs of even order. One reas...
Edge Coloring, Polyhedra and Probability
, 1998
"... Submitted inpartial ful llment of the requirements ..."
Local Conditions for EdgeColouring of Cographs
, 1998
"... In this note, we investigate three versions of the overfull property for colouring graphs and their relation to cographs. Each of these properties implies that the graph cannot be edgecolored with \Delta colors, where \Delta is the maximum degree. Consequently, the graph is a Class 2 graph. These t ..."
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In this note, we investigate three versions of the overfull property for colouring graphs and their relation to cographs. Each of these properties implies that the graph cannot be edgecolored with \Delta colors, where \Delta is the maximum degree. Consequently, the graph is a Class 2 graph. These three versions are not equivalent for general graphs. However, some equivalences hold for the classes of indifference graphs, split graphs, and complete multipartite graphs. Hilton and Chetwynd conjectured that being Class 2 is equivalent to being subgraphoverfull, when the graph has maximum degree greater than 1 3 jV (G)j. A cograph is a graph with no induced subgraph isomorphic to a P4 . We show that connected cographs have maximum degree greater than 1 2 jV (G)j. In view of Hilton and Chetwynd's conjecture, it is interesting to study when a cograph is subgraphoverfull. We prove that for a special class of cographs, obtained by restricting the number of levels in the modular decomposi...
Local Conditions for EdgeColoring
, 1995
"... In this note, we investigate three versions of the overfull property for graphs and their relation to the edgecoloring problem. Each of these properties implies that the graph cannot be edgecolored with \Delta colors, where \Delta is the maximum degree. The three versions are not equivalent for ge ..."
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In this note, we investigate three versions of the overfull property for graphs and their relation to the edgecoloring problem. Each of these properties implies that the graph cannot be edgecolored with \Delta colors, where \Delta is the maximum degree. The three versions are not equivalent for general graphs. However, we show that some equivalences hold for the classes of indifference graphs, split graphs, and complete multipartite graphs. 1991 Mathematics Subject Classification: 05C15, 05C85, 68Q25. 1 Introduction An important problem in graph theory is edgecoloring : coloring the edges of a graph so that incident edges get different colors. The goal is to use the minimum number of colors. A celebrated theorem by Vizing [13, 9] states that this minimum is always \Delta or \Delta +1, where \Delta is the maximum degree of the graph. To decide between these two possibilities is however NPhard [8, 1]. More precisely, if we denote by C1 the class of graphs that are edgecolorable wit...
DROP COST AND WAVELENGTH OPTIMAL TWOPERIOD GROOMING WITH RATIO 4
"... Abstract. We study grooming for twoperiod optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the twoperiod grooming problem, during the first period of time there is alltoall uniform traffic among n nodes, eac ..."
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Abstract. We study grooming for twoperiod optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the twoperiod grooming problem, during the first period of time there is alltoall uniform traffic among n nodes, each request using 1/C of the bandwidth; and during the second period, there is alltoall uniform traffic only among a subset V of v nodes, each request now being allowed to use 1/C ′ of the bandwidth, where C ′ < C. We determine the minimum drop cost (minimum number of ADMs) for any n, v and C = 4 and C ′ ∈ {1, 2, 3}. To do this, we use tools of graph decompositions. Indeed the twoperiod grooming problem corresponds to minimizing the total number of vertices in a partition of the edges of the complete graph Kn into subgraphs, where each subgraph has at most C edges and where furthermore it contains at most C ′ edges of the complete graph on v specified vertices. Subject to the condition that the twoperiod grooming has the least drop cost, the minimum number of wavelengths required is also determined in each case. Key words. theory. traffic grooming, SONET ADM, optical networks, graph decomposition, design
A Comparative Study of Brazilian Beers
, 2004
"... We discuss the following conjecture:If G is a graph with n vertices and maximum degree \Delta> n/3, then G is 1factorizable. ..."
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We discuss the following conjecture:If G is a graph with n vertices and maximum degree \Delta> n/3, then G is 1factorizable.
On the EdgeColoring of Split Graphs
, 1996
"... We consider the following question: can split graphs with odd maximum degree be edgecoloured with maximum degree colours? We show that any odd maximum degree split graph can be transformed into a special split graph. For this special split graph, we were able to solve the question, in case the grap ..."
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We consider the following question: can split graphs with odd maximum degree be edgecoloured with maximum degree colours? We show that any odd maximum degree split graph can be transformed into a special split graph. For this special split graph, we were able to solve the question, in case the graph has a quasiuniversal vertex.
Gunnar Brinkmann
, 2003
"... A graph is chromaticindexcritical if it cannot be edgecoloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest k ..."
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A graph is chromaticindexcritical if it cannot be edgecoloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order 18 [18, 31]. In this paper we show that there are no chromaticindexcritical graphs of order 14. Our result extends that of [5] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexample given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13. Key words: critical graph, edgecolouring, graph generation. Math. Subj. Class (2001): 05C15, 05C30