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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...
COMBINATORICA Bolyai Society – SpringerVerlag Combinatorica 27 (1) (2007) 35–69
, 2004
"... We prove – for sufficiently large n – the following conjecture of Faudree and Schelp: R(Pn,Pn,Pn) = j 2n − 1 for odd n, 2n − 2 for even n, for the threecolor Ramsey numbers of paths on n vertices. 1. ..."
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We prove – for sufficiently large n – the following conjecture of Faudree and Schelp: R(Pn,Pn,Pn) = j 2n − 1 for odd n, 2n − 2 for even n, for the threecolor Ramsey numbers of paths on n vertices. 1.