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How to find overfull subgraphs in graphs with large maximum degree, II
 Discrete Applied Math
, 2000
"... Let G be a simple graph with 3#(G) > V .TheOverfull Graph Conjecture states that the chromatic index of G is equal to #(G), if G does not contain an induced overfull subgraph H with #(H)=#(G), and otherwise it is equal to #(G) + 1. We present an algorithm that determines these subgraphs in O(n 5/3 ..."
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Cited by 6 (0 self)
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Let G be a simple graph with 3#(G) > V .TheOverfull Graph Conjecture states that the chromatic index of G is equal to #(G), if G does not contain an induced overfull subgraph H with #(H)=#(G), and otherwise it is equal to #(G) + 1. We present an algorithm that determines these subgraphs in O(n 5/3 m) time, in general, and in O(n 3 ) time, if G is regular. Moreover, it is shown that G can have at most three of these subgraphs. If 2#(G) #V ,thenG contains at most one of these subgraphs, and our former algorithm for this situation is improved to run in linear time. 1
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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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...
Minimum degree conditions for the Overfull Conjecture for odd order graphs
"... The Overfull Conjecture states that a graph G with 3∆(G) ≥ n(G) is Class 2 if and only if it has a ∆(G)overfull subgraph. M. Plantholt showed that the Overfull Conjecture is true for graphs with even order and high minimum degree. In this paper we look at graphs with odd order and show under which ..."
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The Overfull Conjecture states that a graph G with 3∆(G) ≥ n(G) is Class 2 if and only if it has a ∆(G)overfull subgraph. M. Plantholt showed that the Overfull Conjecture is true for graphs with even order and high minimum degree. In this paper we look at graphs with odd order and show under which restrictions the Overfull Conjecture holds true for these. 1
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