Polygonal graphs

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-claw of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal.

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.

References

M. Perkel, Bounding the valency of polygonal graphs with odd girth, Can. J. Math. 31 (1979) 1307-1321.

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.