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Steiner Triple Systems, Pinched Surfaces, and Complete Multigraphs

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Abstract

We consider complete multigraphs \({K_n^m}\) on n vertices with every pair joined by m edges. We embed these graphs to triangulate \({S_n^k}\) , a pinched surface with n pinch points each having k sheets. These embeddings have a vertex at each pinch point and any two sheets at a pinch point have the same number of edges. Moreover, we want to 2m-color the faces such that each color class is a Steiner triple system. These embeddings generalize in two ways biembeddings of Steiner triple systems, the case m =  1, k =  1 of simple graphs in surfaces without pinch points.

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  1. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, 05405, USA

    Dan Archdeacon

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  1. Dan Archdeacon
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Correspondence to Dan Archdeacon.

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Archdeacon, D. Steiner Triple Systems, Pinched Surfaces, and Complete Multigraphs. Graphs and Combinatorics 30, 1351–1361 (2014). https://doi.org/10.1007/s00373-013-1348-2

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  • DOI: https://doi.org/10.1007/s00373-013-1348-2

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