On the Hamilton–Waterloo Problem with cycle lengths of distinct parities

Let $K_v^1$ denote the complete graph $K_v$ if $v$ is odd and $K_v?I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v,M,N,\alpha ,\beta$, the Hamilton–Waterloo problem asks for a 2-factorization of $K_v^1$ into $\alpha$$C_M$-factors and $\beta$$C_N$-factors. Clearly, $M,N\ge 3$, $M\mid v$, $N\mid v$ and $\alpha +\beta =?\frac{v?1}{2}?$ are necessary conditions.Very little is known on the case where $M$ and $N$ have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever $M|N$, $v>6N>36M$, and $\beta \ge 3$.     

Resolvable gregarious cycle decompositions of complete equipartite graphs

The complete multipartite graph K_n(m) with n parts of size m is shown to have a decomposition into n-cycles in such a way that each cycle meets each part of K_n(m); that is, each cycle is said to be gregarious. Furthermore, gregarious decompositions are given which are also resolvable.     

A construction of non self-orthogonal designs

It is known that a self-orthogonal 2-(21,6,4) design is unique up to isomorphism. We give a construction of 2-(21,6,4) designs. As an example, we obtain non self-orthogonal 2-(21,6,4) designs. Furthermore, we also consider a generalization of the construction.     

Tripling Construction for Large Sets of Resolvable Directed Triple Systems

In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.     

Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

We first define a transitive resolvable idempotent quasigroup (TRIQ), and show that a TRIQ of order v exists if and only if 3∣v and . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems s, which improves an earlier version of tripling construction by Kang. As an application we obtain an for any integer n≥1, which provides an infinite family of even orders.     

Resolvable even cycle decompositions of the tensor product of complete graphs

In this paper, we consider resolvable k-cycle decompositions (for short, k-RCD) of K_m×K_n, where × denotes the tensor product of graphs. It has been proved that the standard necessary conditions for the existence of a k-RCD of K_m×K_n are sufficient when k is even.