A class 1 graph is a simple graph whose edge chromatic number equals its maximum vertex
degree 
. Vizing's theorem
states that a simple 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 
.
The 
king
graph is class 1 for all positive integers 
and 
.
The 
-windmill
graph, formed from 
copies of the complete graph 
with one shared graph vertex,
is class 1 for integers 
(S. Wagon, pers. comm., Oct. 27-28, 2011).
Keller graphs are class 1 (Jarnicki et al. 2017).
For positive integers 
and 
, rook
graphs 
are class 1 when 
is even, and class 2
when 
is odd and greater than 1. The singleton
graph is class 1.
The numbers of nonisomorphic class 1 simple graphs on 
, 2, ... vertices
are 1, 2, 3, 10, 28, 145, ... (OEIS A099435).
Similarly, the numbers of nonisomorphic connected class 1 simple graphs are 1, 1, 1, 6, 17, 109, 821, 11050, 260150, ... (OEIS A099436; Royle).
en cycles hamiltoniens."
Disc. Math. 37, 19-27, 1981.