RESOLVABLE MENDELSOHN DESIGNS AND FINITE FROBENIUS GROUPS

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v,k,1)$-Mendelsohn design for any integers $v>k\geq 2$ with $v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\unicode[STIX]{x1D719}$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K\rtimes \langle \unicode[STIX]{x1D719}\rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}\geq 3$ in prime factorisation and any prime $k$ dividing $p_{i}^{e_{i}}-1$ for $1\leq i\leq t$, there exists a resolvable perfect $(v,k,1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i}-1$ for $1\leq i\leq t$, then there are at least $\unicode[STIX]{x1D711}(k)^{t}$ resolvable $(v,k,1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\unicode[STIX]{x1D711}$ is Euler’s totient function.

Holey Perfect Mendelsohn Designs of Type 2nu1 with Block Size Four

Let 4-HPMD denote a holey perfect Mendelsohn design with block size four. The existence of 4-HPMDs with n holes of size 2 and one hole of size 3, that is, of type 2n31, was established by Bennett et al. in 1997. In this paper, we investigate the existence of 4-HPMDs of type 2nu1 for 1≤u≤16: a 4-HPMD(2nu1) exists if and only if n≥max(4,u+1), except possibly for (n,u)=(7,5), (7, 6), (11, 9), (11, 10). We also investigate the existence of 4-HPMD(2nu1) for general u and prove that there exists a 4-HPMD(2nu1) for all n≥⌈5u/4⌉+4. Moreover, if u≥35, then a 4-HPMD(2nu1) exists for all n≥⌈5u/4⌉+1; if u≥95, then a 4-HPMD(2nu1) exists for all n≥⌈5u/4⌉−2.