All generalized Petersen graphs are unit-distance graphs

Abstract

In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of I-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each I-graph I(n; j; k) admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every I-graph I(n; j; k) has an isomorphic I-graph that admits a unit-distance representation in the Euclidean plane with a n-fold rotational symmetry, with the exception of the families I(n; j; j) and I(12m; m; 5m), m ≥ 1. We also provide unit-distance representations for these graphs. © 2012 The Korean Mathematical Society.