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Korean Mathematical Society
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ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2012; 49(3): 475-491

Printed May 1, 2012

https://doi.org/10.4134/JKMS.2012.49.3.475

Copyright © The Korean Mathematical Society.

All generalized Petersen graphs are unit-distance graphs

Arjana \v{Z}itnik, Boris Horvat, and Toma\v{z} Pisanski

University of Ljubljana, University of Ljubljana, University of Ljubljana and University of Primorska

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 $I(12m,m, 5m)$, $m \ge 1$. We also provide unit-distance representations for these graphs.

Keywords: unit-distance graph, $I$-graph, generalized Petersen graph, graph representation, degenerate representation, graph isomorphism

MSC numbers: 52C10, 05C10, 05C62, 11A99, 11Z05, 51A20, 52C30, 68R10, 05E18

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