Class 1 Graph

A class 1 graph is a graph whose edge chromatic number equals its maximum vertex degree . Vizing's theorem states that a graph can be edge-colored in either or colors.

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).