An asymptotic solution to the cycle decomposition problem for complete graphs

Let m₁,m₂,…,mt be a list of integers. It is shown that there exists an integer N such that for all n?N, the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m₁,m₂,…,mt if and only if n is odd, 3?mi?n for i=1,2,…,t, and . In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching.     

Packing cycles in complete graphs

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.     

On Minimal Triangulations of Products of Convex Polygons

We give new lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving the lower bounds in Bowen et al. (Topology 44:321–339, 2005).     

Planar graphs decomposable into a forest and a matching

He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching.     

Balls,bins, and embeddings of partial k-star designs

Define a $k$-star to be the complete bipartite graph $K_{1,k}$. In a 2014 article, Hoffman and Roberts prove that a partial $k$-star decomposition of $K_n$ can be embedded in a $k$-star decomposition of $K_{n+s}$ where $s$ is at most $7k?4$ if $k$ is odd and $8k?4$ if $k$ is even. In our work, we offer a straightforward construction for embedding partial $k$-star designs and lower these bounds to $3k?2$ and $4k?2$, respectively.     

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.