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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...
Chromaticindexcritical graphs of order 13 and 14
, 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.
ChromaticIndex Critical Multigraphs of Order 20
, 2000
"... A multigraph M with maximum degree \Delta(M ) is called critical, if the chromatic index Ø 0 (M) ? \Delta(M ) and Ø 0 (M \Gamma e) = Ø 0 (M) \Gamma 1 for each edge e of M . The weak critical graph conjecture [1, 7] claims that there exists a constant c ? 0 such that every critical multigraph M ..."
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A multigraph M with maximum degree \Delta(M ) is called critical, if the chromatic index Ø 0 (M) ? \Delta(M ) and Ø 0 (M \Gamma e) = Ø 0 (M) \Gamma 1 for each edge e of M . The weak critical graph conjecture [1, 7] claims that there exists a constant c ? 0 such that every critical multigraph M with at most c \Delta \Delta(M ) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum degree k for all k 5.