Class 1 Graph

Vizing's theorem states that a graph can be edge-colored in either Delta or Delta+1 colors, where Delta is the maximum vertex degree of the graph. A graph with edge chromatic number equal to Delta is known as a class 1 graph.

König's line coloring theorem states that all bipartite graphs are class 1. All cubic Hamiltonian graphs are class 1, as are planar graphs with maximum vertex degree Delta>7 (Cole and Kowalik 2008).

Class 1 graphs have both edge chromatic number and fractional edge chromatic number equal to Delta.

Families of non-bipartite graphs that appear to be class 1 (or at least whose smallest members are all class 1) include king, Lindgren-Sousselier, and windmill graphs (S. Wagon, pers. comm., Oct. 27-28, 2011). Keller graphs are class 1 (Jarnicki et al. 2015). Aubert and Schneider (1982) showed that rook graphs admit Hamiltonian decomposition, meaning they are class 1 when they have even vertex count and class 2 when they have odd vertex count (because they are odd regular).


The numbers of simple class 1 graphs on n=1, 2, ... nodes are 1, 2, 3, 10, 28, 145, ... (OEIS A099435).


Similarly, the numbers of simple connected class 1 graphs are 1, 1, 1, 6, 17, 109, 821, 11050, 260150, ... (OEIS A099436; Royle).

See also

Class 2 Graph, Edge Chromatic Number, König's Line Coloring Theorem, Vizing's Theorem

Portions of this entry contributed by Ed Pegg, Jr. (author's link)

Portions of this entry contributed by Stan Wagon

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Aubert, J. and Schneider, B. "Décomposition de la somme cartésienne d'un cycle et de l'union de deux cycles hamiltoniens en cycles hamiltoniens." Disc. Math. 38, 7-16, 1982.Cole, R. and Kowalik, L. "New Linear-Time Algorithms for Edge-Coloring Planar Graphs." Algorithmica 50, 351-368, 2008.Jarnicki, W.; Myrvold, W.; Saltzman, P.; and Wagon, S. "Properties, Proved and Conjectured, of Keller, Mycielski, and Queen Graphs." To appear in Ars Math. Comtemp. 2017.Royle, G. "Class 2 Graphs.", N. J. A. Sequences A099435 and A099436 in "The On-Line Encyclopedia of Integer Sequences."

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Class 1 Graph

Cite this as:

Pegg, Ed Jr.; Wagon, Stan; and Weisstein, Eric W. "Class 1 Graph." From MathWorld--A Wolfram Web Resource.

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