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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...
Classifications and Characterisations of Snarks
, 1994
"... This paper considers vertexreductions of snarks. Each reduction naturally divides the class of snarks into three classes. The first class contains those snarks which are not reducible to a snark. The second contains those which are not reducible to a 3colorable graph and the third one consists of ..."
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This paper considers vertexreductions of snarks. Each reduction naturally divides the class of snarks into three classes. The first class contains those snarks which are not reducible to a snark. The second contains those which are not reducible to a 3colorable graph and the third one consists of the snarks which are reducible to a snark as well as to a 3colorable graph. We characterise all these classes in terms of 2factors. Furthermore we characterise snarks which are divided into 3colorable components by the removal of any edge cutset in terms of 3critical subgraphs and show that there are infinitely many such snarks. 1 Introduction We follow the terminology and notation of Bondy and Murty [2]. We consider proper edge colorings of 3regular graphs. The set of 3regular graphs is divided into two classes by a famous theorem of Vizing which states that each simple loopless graph is edgecolorable with \Delta or \Delta +1 colors, see e.g. [2]. Class I contains the 3colorable gr...