Related papers: A method to construct generalized balanced tournam…
There is a one-to-one correspondence between the point set of a group divisible design (GDD) with $v_1$ groups of $v_2$ points and the edge set of a complete bipartite graph $K_{v_1,v_2}$. A block of GDD corresponds to a subgraph of…
A $(v,k,\lambda)$-BIBD $(X,\mathcal B)$ can be nested if there is a mapping $\phi:\mathcal B\rightarrow X$ such that $(X,\{B\cup\{\phi(B)\}\mid B\in\mathcal B\})$ is a $(v,k+1,\lambda+1)$-packing. A $(v,k,\lambda)$-BIBD has a (perfect)…
A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erd\H{o}s and J. W. Moon. By probabilistic methods, the existence of "unrankable" tournaments was proved. On the other hand, they…
A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph,…
Balanced incomplete block designs (BIBDs) have wide applications in engineering, business and sciences. In this paper, for each (v, k, \lambda)-BIBD, we construct a strongly symmetric k-th order v-dimensional tensor. We call such a strongly…
For two integers $n\geq 3$ and $2\leq p\leq n$, we denote $D(n,p)$ the digraph obtained from a directed $n$-cycle by changing the orientations of $p-1$ consecutive arcs. In this paper, we show that a family of $k$-regular $(k\geq 3)$…
Every sport needs rules. Tournament design refers to the rules that determine how a tournament, a series of games between a number of competitors, is organized. This study aims to provide an overview of the tournament design literature from…
A $k$-regular graph on $v$ vertices is a {\em divisible design graph} if there exist integers $\lambda_1,\lambda_2,m,n$ such that the vertex set can be partitioned into $m$ classes of size $n$ and any two different vertices from the same…
We prove that a 3-GDD of type $1^n k^1 \ell^1$, where $n= k \cdot \ell$, with minimum distance 3 exists for every $k$ and $\ell$ such that $n = k \ell$, $k = 1$ or $3~(mod ~ 6)$, and $\ell = 1$ or $3~(mod ~ 6)$. These designs are of the…
It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection…
K-geodetic graphs (K capital) are defined as graphs in which each pair of nonadjacent vertices has at most K paths of minimum length between them. A K-geodetic graph is geodetic if K=1, bigeodetic if K=2 and trigeodetic if K=3. K-geodetic…
The bipartite traveling tournament problem (BTTP) addresses inter-league sports scheduling, which aims to design a feasible bipartite tournament between two $n$-team leagues under some constraints such that the total traveling distance of…
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…
Complex orthogonal designs (CODs) play a crucial role in the construction of space-time block codes. Their real analog, real orthogonal designs (or equivalently, sum of squares composition formula) have a long history. Adams et al. (2011)…
We give explicit constructions for incomplete pairwise balanced designs IPBD$((v;w),K)$, or, equivalently, edge-decompositions of a difference of two cliques $K_v \setminus K_w$ into cliques whose sizes belong to the set $K$. Our…
A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph,…
Complex orthogonal design (COD) with parameter $[p, n, k]$ is a combinatorial design used in space-time block codes (STBCs). For STBC, $n$ is the number of antennas, $k/p$ is the rate, and $p$ is the decoding delay. A class of rate $1/2$…
Let $TT_k$ denote the transitive tournament on $k$ vertices. Let $TT(h,k)$ denote the graph obtained from $TT_k$ by replacing each vertex with an independent set of size $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$…
A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of…
A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a matrix with $0$s on its diagonal and all other entries equal to $1$. An outstanding…