English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we will show the existence of a face $2$-colourable biembeddingof the complete graph onto an orientable surface where each face is a cycle ofa fixed length $k$, for infinitely many values of $k$. In particular, undercertain conditions, we show that there exists at least $(n-2)[(p-2)!]^2/(e^2kn)$ non-isomorphic face $2$-colourable biembeddings of $K_{2nk+1}$ in whichall faces are cycles of length $k=4p+3$. These conditions are: $n\equiv 1\mod4$, $k\equiv 3\mod 4$ and either $n$ is prime or $n\gg k$ and $n\equiv 0\mod 3$implies $p\equiv 1\mod 3$. To achieve this result we begin by verifying theexistence of $(n-2)[(p-2)!/e]^2$ non-equivalent Heffter arrays, $H(n;k)$, whichsatisfy the conditions: (1) for each row and each column the sequential partialsums determined by the natural ordering must be distinct modulo $2nk+1$; (2)the composition of the natural orderings of the rows and columns is equivalentto a single cycle permutation on the entries in the array. The existence ofHeffter arrays $H(n;k)$ that satisfy condition (1) was established earlier in\cite{BCDY} and in this current paper we vary this construction and show thatthere are at least $(n-2)[(p-2)!/e]^2$ such non-equivalent $H(n;k)$ thatsatisfy condition (1) and then show that each of these Heffter arrays alsosatisfy condition (2) under certain conditions.

Biembeddings of cycle systems using integer Heffter arrays