Class 1 Graph

Vizing's theorem states that a graph can be edge-colored in either  or  colors, where  is the maximum vertex degree of the graph. A graph with edge chromatic number equal to  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  (Cole and Kowalik 2008).

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

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 , 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).

REFERENCES

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." http://school.maths.uwa.edu.au/~gordon/remote/graphs/#class2.Sloane, N. J. A. Sequences A099435 and A099436 in "The On-Line Encyclopedia of Integer Sequences."