ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS

Title & Authors
ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS
Zitnik, Arjana; Horvat, Boris; Pisanski, Tomaz;

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 $\small{I}$-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each $\small{I}$-graph $\small{I(n,j,k)}$ 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 $\small{I}$-graph $\small{I(n,j,k)}$ has an isomorphic $\small{I}$-graph that admits a unit-distance representation in the Euclidean plane with a $\small{n}$-fold rotational symmetry, with the exception of the families $\small{I(n,j,j)}$ and $\small{I(12m,m,5m)}$, $\small{m{\geq}1}$. We also provide unit-distance representations for these graphs.
Keywords
unit-distance graph;I-graph;generalized Petersen graph;graph representation;degenerate representation;graph isomorphism;
Language
English
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