On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs [article]

Simone Costa, Anita Pasotti
2022 arXiv   pre-print
This article aims to provide exponential lower bounds on the number of non-isomorphic k-gonal biembeddings of the complete multipartite graph into orientable surfaces. For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays
more » ... ted by Cavenagh, Donovan and Yazici in 2020, we obtain that, for infinitely many values of k and v, there are at least k^k/2+o(k)· 2^v·H(1/4)/(2k)^2+o(v) non-isomorphic k-gonal biembeddings of K_v, where H(·) is the binary entropy. Moreover about the embeddings of K_v/t× t, for t∈{1,2,k}, we provide a construction of 2^v·H(1/4)/2k(k-1)+o(v,k) non-isomorphic k-gonal biembeddings whenever k is odd and v belongs to a wide infinite family of values.
arXiv:2111.08323v2 fatcat:tqyfb4pb45ehrcv63pibjjbtjq