Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

Abstract

We construct a face two‐colourable, blue and green say, embedding of the complete graph Kn in a nonorientable surface in which there are (n−1)/2 blue faces each of which have a hamilton cycle as their facial walk and n(n−1)/6 green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of Kn, for all n≡3(mod36) and n≠3. Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph K36k+3+Km¯, for m=18k+1+6s where k≥1 and 0≤s≤k−1.

Globally simple Heffter arrays H(n;k) when k≡0Burrage1, Donovan2, Cavenagh3 et al. 2020Discrete Mathematics181410View full textAdd to dashboardCite

“…Since then a number of papers have appeared on the connection between Heffter arrays and the biembedding of cycle decompositions as well as a number of papers studying more general biembeddings of the complete graph. See for examples the papers [2,5,11,12,14,16,17,18,21,23] and [10].…”

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

“…The nonorientable genus was also determined by Jungerman [11]. The genus of graphs in the family K m + K n has been investigated in a number of articles, including [2,5,9,13,15]. The work here extends that of the first author and Stephens in [6].…”

“…While preparing the final revision of this article we were given details of an unpublished construction by JozefŠiráň for an orientable hamilton cycle embedding of K 15 , which involves gluing together three embeddings derived from embedded voltage graphs. T (15) covers new cases m = 15, 42, 82, 123, 162, 243, 322, 366, 483, . .…”

“…Proposition 5.10. Suppose there exists an orientable hamilton cycle embedding of K 15 and of K n for all n ≥ 11 such that n ≡ 7, 11,14,19,22 or 23 (mod 24). Then there exists an orientable hamilton cycle embedding of K n for all n ≡ 2 or 3 (mod 4), n ∈ {2, 6, 7}.…”

Biembeddings of cycle systems using integer Heffter arrays

“…In 2016 Griggs and McCourt [19] developed new constructions for biembeddings of symmetric () ‐cycle decompositions of and established necessary and sufficient conditions for the existence of a biembedding of symmetric ‐cycle systems on a nonorientable surface if and on an orientable surface if is odd and . Other studies connecting cycle decompositions and embeddings on orientable and nonorientable surfaces include [1,8,11,13,16,18,19,23,24] and [10].…”