Some remarks on a paper by Vizing on critical graphs

Fiorini, Stanley

doi:10.1017/s030500410005129x

Cited by 27 publications

(14 citation statements)

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“…I know of only four articles that obtain results that are valid for arbitrarily large values of . For all values of 2, Fiorini [1] proved that d 1 2 ( + 1) ( odd);d 1 2 ( + 2) ( even).…”

Section: Introductionmentioning

confidence: 99%

The average degree of an edge‐chromatic critical graph II

Woodall

2007

Journal of Graph Theory

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“…I know of only four articles that obtain results that are valid for arbitrarily large values of . For all values of 2, Fiorini [1] proved that d 1 2 ( + 1) ( odd);d 1 2 ( + 2) ( even).…”

Section: Introductionmentioning

confidence: 99%

The average degree of an edge‐chromatic critical graph II

Woodall

2007

Journal of Graph Theory

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“…Other similar constructions not related to the Petersen graph are to be found in [14,52,57,60,102,1411. Most works of a general nature, such as [8,9,11,15,82,165,168, 1711 include a treatment of bipartite graphs and of complete graphs.…”

Section: Classificationmentioning

confidence: 94%

“…Various generalizations of the Petersen graph have been considered [21,22,34,35,37,67,95,106,122,123,162,163,1701, including methods of constructing class-two graphs. Other similar constructions not related to the Petersen graph are to be found in [14,52,57,60,102,1411. Most works of a general nature, such as [8,9,11,15,82,165,168, 1711 include a treatment of bipartite graphs and of complete graphs.…”

Section: Classificationmentioning

confidence: 99%

“…Bounds on the number of edges of critical graphs are found in [14,55,57,60,61,160,1611 whereas circuit length properties in connection with the girth (length of shortest circuit) and the circumference (length of longest circuit) are found in [52,58,1121 and in [57, 1601, respectively. Hamiltonian graphs are discussed in [54, 721. Using properties of critical graphs, it is shown in [50] that almost all graphs are of class one. Other applications of critical graphs have been made in the investigation of graphs with particular topological properties [74,120,121,.…”

Section: Applicationmentioning

confidence: 99%

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A bibliographic survey of edge‐colorings

Fiorini

1978

Journal of Graph Theory

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This paper presents a bibliography on edge-colorings of graphs which is as complete as possible to date. W e introduce the papers by a brief discussion of the ideas involved.All graphs we consider will contain no loops but may have multiple edges. We call graphs without multiple edges simple. An edge-coloring of a graph is an assignment of colors to the edges in such a way that adjacent edges receive different colors. The least number of colors required is called the chromatic class of the (mu1ti)graph. If the graph is simple, then this number is called the chromatic index (or the edge chromatic number).Edge-colorings of graphs seem to have been first considered in 1880 by P. G. Tait [142] who showed the equivalence of the four color problem to that of showing that a cubic, bridgeless simple graph has chromatic index 3. Notable contributions to the theory came from D. Konig [109,110] in 1916, who showed that bipartite graphs have chromatic class equal to their maximum valemy,-and from C. E. Shannon [135] in 1949 who showed that an upper bound for the chromatic class is [Zp] where p denotes the maximum valency.Apart from these, little was done until 1964 when V. G. Vizing [156] showed that the chromatic class of a graph of maximum valency p and maximum multiplicity of edges p is at most p + p . Most of the early papers, prior to 1964, were concerned with the relationship between edge-colorings and vertex-colorings.

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“…It was proved in 1975 by Fiorini [2] that if G is a D-critical graph, then jEj5 1 4 jV jðD þ 1Þ: 1 To whom correspondence should be addressed. 2 Research partially supported by the National Science Foundation Grants DMS-9896292 and DMS 0096160. More than 20 years later, Fiorini's result was improved by two recent papers which are summarized by the following theorem.…”

Section: Introductionmentioning

confidence: 99%