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http://dx.doi.org/10.4134/JKMS.2012.49.3.475

ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS  

Zitnik, Arjana (IMFM University of Ljubljana)
Horvat, Boris (IMFM University of Ljubljana)
Pisanski, Tomaz (IMFM University of Ljubljana and University of Primorska)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.3, 2012 , pp. 475-491 More about this Journal
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 $I(12m,m,5m)$, $m{\geq}1$. We also provide unit-distance representations for these graphs.
Keywords
unit-distance graph; I-graph; generalized Petersen graph; graph representation; degenerate representation; graph isomorphism;
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