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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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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
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 NPhard. We pose a n ..."
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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 NPhard. 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 rgraphs.
Independent Sets and 2Factors in EdgeChromaticCritical Graphs
"... In 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs ..."
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In 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs with many edges, and determine upper bounds for the size of independent vertex sets in those graphs. 1 Introduction In this paper we consider simple graphs. Beside our definitions we use standard graph theoretical notation. Vizing showed that the chromatic index 0 (G) of a graph G equals its maximum vertex degree \Delta(G) or \Delta(G) + 1. A graph G is \Deltacritical if it has chromatic index \Delta +1 and each proper subgraph of G has smaller chromatic index. If the value of \Delta is unimportant we use critical graph instead of \Deltacritical. 1 A graph G = (V; E) with more than \Deltab jV j 2 c edges cannot be colored with \Delta colors, since it has too many edges. Such grap...