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Regular graphs of high degree are 1-factorizable
- Proc. London Math. Soc
, 1985
"... It is a well-known conjecture that if a regular graph G of order 2n has degree d(G) satisfying d(G) ^ n, then G is the union of edge-disjoint 1-factors. It is well known that this conjecture is true for d(G) equal to 2n —1 or 2n — 2. We show here that it is true for d(G) equal to2n — 3, In — 4, or2 ..."
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Cited by 12 (4 self)
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It is a well-known conjecture that if a regular graph G of order 2n has degree d(G) satisfying d(G) ^ n, then G is the union of edge-disjoint 1-factors. It is well known that this conjecture is true for d(G) equal to 2n —1 or 2n — 2. We show here that it is true for d(G) equal to2n — 3, In — 4, or2n — 5. We also show that it is true for </(G)>$|K(G)|. 1.
Chromatic Index Critical Graphs of Even Order with Five Major Vertices
"... We prove that there does not exist any chromatic index critical graph of even order with exactly five vertices of maximum degree. This extends an earlier result of Chetwynd and Hilton who proved the same with five replaced by four or three. 1 ..."
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Cited by 4 (0 self)
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We prove that there does not exist any chromatic index critical graph of even order with exactly five vertices of maximum degree. This extends an earlier result of Chetwynd and Hilton who proved the same with five replaced by four or three. 1
Chromatic Index Critical Graphs of Orders 11 and 12
, 1997
"... A chromatic-index-critical graph G on n vertices is non-trivial if it has at most \Deltab n 2 c edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial 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 chromatic-index-critical graph G on n vertices is non-trivial if it has at most \Deltab n 2 c edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial 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...
Excessive Factorizations of Regular Graphs
"... Abstract. An excessive factorization of a graph G is a minimum set F of 1–factors of G whose union is E(G). In this paper we study excessive factorizations of regular graphs. We introduce two graph parameters related to excessive factorizations and show that their computation is NP-hard. We pose a n ..."
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Cited by 3 (1 self)
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Abstract. An excessive factorization of a graph G is a minimum set F of 1–factors of G whose union is E(G). In this paper we study excessive factorizations of regular graphs. We introduce two graph parameters related to excessive factorizations and show that their computation is NP-hard. We pose a number of questions regarding these parameters. We show that the size of an excessive factorization of a regular graph can exceed the degree of the graph by an arbitrarily large quantity. We conclude with a conjecture on the excessive factorizations of r-graphs.
