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).

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."
Algorithmica50, 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."