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      저자 : Pisanski, Tomaz (검색결과 2 건)
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        ALL GENERALIZED PETERSEN GRAPHS ARE UNIT-DISTANCE GRAPHS

        Zitnik, Arjana,Horvat, Boris,Pisanski, Tomaz Korean Mathematical Society 2012 대한수학회지 Vol.49 No.3

        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.

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        All generalized Petersen graphs are unit-distance graphs

        Arjana Zitnik,Boris Horvat,Tomaz Pisanski 대한수학회 2012 대한수학회지 Vol.49 No.3

        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 ≥ 1. We also provide unit-distance representations for these graphs. 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 ≥ 1. We also provide unit-distance representations for these graphs.

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