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  • Distinguishing partitions and asymmetric uniform hypergraphs
    Ellingham, Mark N. ; Schroeder, Justin Z.
    A distinguishing partition for an action of a group ▫$\Gamma$▫ on a set ▫$X$▫ is a partition of ▫$X$▫ that is preserved by no nontrivial element of ▫$\Gamma$▫. As a special case, a distinguishing ... partition of a graph is a partition of the vertex set that is preserved by no nontrivial automorphism. In this paper we provide a link between distinguishing partitions of complete equipartite graphs and asymmetric uniform hypergraphs. Suppose that ▫$m \geq 1$▫ and ▫$n \geq 2$▫. We show that an asymmetric ▫$n$▫-uniform hypergraph with ▫$m$▫ edges exists if and only if ▫$m \geq f(n)$▫, where ▫$f(2) = f(14) = 6, f(6) = 5$▫, and ▫$f(n) = \lfloor \log_2 (n + 1) + 2 \rfloor$▫ otherwise. It follows that a distinguishing partition of ▫$K_{m(n)} = K_{n,n, \dots ,n}$▫, or equivalently for the wreath product action ▫$S_n$▫ Wr ▫$S_m$▫, exists if and only if ▫$m \geq f(n)$▫.
    Vir: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 4, no. 1, 2011, str. 111-123)
    Vrsta gradiva - članek, sestavni del
    Leto - 2011
    Jezik - angleški
    COBISS.SI-ID - 16264793

    Povezava(-e):

    http://amc.imfm.si/index.php/amc/article/view/195
    Digitalna knjižnica Slovenije - dLib.si

vir: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 4, no. 1, 2011, str. 111-123)

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