2019/06/23 by Cavenagh, Nicholas J., Donovan, D., Yazici, E. Ş. · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1906.10525
In this paper we will show the existence of a face 2-colourable biembedding of the complete graph onto an orientable surface where each face is a cycle of a fixed length k, for infinitely many values of k. In particular, under certain conditions, we show that there exists at least (n-2)[(p-2)!]2/(e2 kn) non-isomorphic face 2-colourable biembeddings of K2nk+1 in which all faces are cycles of length k=4p+3. These conditions are: n≡ 1\mod 4, k≡ 3\mod 4 and either n is prime or n≫ k and n≡ 0\mod 3 implies p≡ 1\mod 3. To achieve this result we begin by verifying the existence of (n-2)[(p-2)!/e]2 non-equivalent Heffter arrays, H(n;k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo 2nk+1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arrays H(n;k) that satisfy condition (1) was established earlier in \citeBCDY and in this current paper we vary this construction and show that there are at least (n-2)[(p-2)!/e]2 such non-equivalent H(n;k) that satisfy condition (1) and then show that each of these Heffter arrays also satisfy condition (2) under certain conditions.