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Journal of the Korean Mathematical Society
Article . 2012 . Peer-reviewed
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ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS

Authors: Arjana Zitnik; Boris Horvat; Tomaz Pisanski;

ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS

Abstract

Abstract. In 1950 a class of generalized Petersen graphs was introducedby Coxeter and around 1970 popularized by Frucht, Graver and Watkins.The family of I-graphs mentioned in 1988 by Bouwer et al. representsa slight further albeit important generalization of the renowned Petersengraph. We show that each I-graph I(n;j;k) admits a unit-distance rep-resentation in the Euclidean plane. This implies that each generalizedPetersen graph admits a unit-distance representation in the Euclideanplane. In particular, we show that every I-graph I(n;j;k) has an isomor-phic I-graph that admits a unit-distance representation in the Euclideanplane with a n-fold rotational symmetry, with the exception of the fam-ilies I(n;j;j) and I(12m;m;5m), m  1. We also provide unit-distancerepresentations for these graphs. 1. IntroductionI-graphs were introduced in the Foster census [5] and form a natural gen-eralization of the generalized Petersen graphs introduced by Coxeter [8] andnamed by Watkins [26]. This well-known family of graphs has been extensivelystudied [1, 10, 18, 20, 22, 25].Let n3 and j;kbe such that 1 j, k

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    popularity
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    influence
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    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
5
Average
Average
Average
gold