id: 06045504
dt: j
an: 06045504
au: Žitnik, Arjana; Horvat, Boris; Pisanski, Tomaž
ti: All generalized Petersen graphs are unit-distance graphs.
so: J. Korean Math. Soc. 49, No. 3, 475-491 (2012).
py: 2012
pu: The Korean Mathematical Society, Seoul
la: EN
cc: 
ut: unit-distance graph; $I$-graph; generalized Petersen graph; graph
    representation; degenerate representation; graph isomorphism
ci: 
li: doi:10.4134/JKMS.2012.49.3.475
ab: Summary: In 1950 a class of generalized Petersen graphs was introduced by
    {\it H.S.M. Coxeter} [Bull. Am. Math. Soc. 56, 413‒455 (1950; Zbl
    0040.22803)] and around 1970 popularized by {\it R. Frucht}, {\it J.E.
    Graver} and {\it M.E. Watkins} [Proc. Camb. Philos. Soc. 70, 211-218
    (1971; Zbl 0221.05069)]. The family of $I$-graphs mentioned in 1988 by
    {\it I.Z. Bouwer} et al. [“The Foster census. R.M. Foster’s census
    of connected symmetric trivalent graphs", Winnipeg (Canada): Charles
    Babbage Research Centre. vii, 240 p. (1988; Zbl 0639.05043)] 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.
rv: