PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

All generalized Petersen graphs are unit-distance graphs
Tomaž Pisanski, Boris Horvat and Arjana Žitnik
Preprint series Volume 48, Number 1109, pp. 1-16, 2010. ISSN 2232-2094

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 Eu- clidean 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 sym- metry, 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.

EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Theory & Algorithms
ID Code:8284
Deposited By:Boris Horvat
Deposited On:21 February 2012