The 1-skeleton of a regular solid is polygonal.
All rectagraphs are 4-gonal graphs; all (0,2)-graphs are near 4-gonal graphs.
The Petersen graph and the Perkel graph and the J1 graph are 5-gonal.
M. Perkel, Near-polygonal graphs, Ars. Comb. 26A (1988) 149-170.
M. Perkel and C.E. Praeger, Polygonal graphs: New families and an approach to their analysis, Congressus Numerantium 124 (1997) 161-173.
M. Perkel, C.E. Praeger & R. Weiss, On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit, J. of Algebraic Combinatorics 13 (2001) 257-273.
Cai Heng Li and Ákos Seress, Symmetrical path-cycle covers of a graph and polygonal graphs, JCT (A) 114 (2007) 35-51.