In this paper, we prove the existence of \(22\) new \(3\)-designs on \(26\) and \(28\) points. The base of the constructions are two designs with a small maximum size of the intersection of any two blocks.

A large set of KTS(\(v\)), denoted by LKTS(\(v\)), is a collection of (\(v-2\)) pairwise disjoint KTS(\(v\)) on the same set. In this article, some new LKTS(\(v\)) is constructed.

Let \(G\) be a graph with \(v\) vertices. If there exists a list of colors \(S_1, S_2, \ldots, S_v\) on its vertices, each of size \(k\), such that there exists a unique proper coloring for \(G\) from this list of colors, then \(G\) is called a uniquely \(k\)-list colorable graph. We prove that a connected graph is uniquely \(2\)-list colorable if and only if at least one of its blocks is not a cycle, a complete graph, or a complete bipartite graph. For each \(k\), a uniquely \(k\)-list colorable graph is introduced.

A supergraph \(H\) of a graph \(G\) is called tree-covered if \(H – E(G)\) consists of exactly \(|V(G)|\) vertex-disjoint trees, with each tree having exactly one point in common with \(G\). In this paper, we show that if a graph \(G\) can be packed in its complement and if \(H\) is a tree-covered supergraph of \(G\), then \(G\) itself is self-packing unless \(H\) happens to be a member of a specified class of graphs. This is a generalization of earlier results that almost all trees and unicyclic graphs can be packed in their complements.

Let \(T = (V,A)\) be an oriented graph with \(n\) vertices. \(T\) is completely strong path-connected if for each arc \((a,b) \in A\) and \(k\) (\(k = 2, \ldots, n-1\)), there is a path from \(b\) to \(a\) of length \(k\) (denoted by \(P_k(a,b)\)) and a path from \(a\) to \(b\) of length \(k\) (denoted by \(P’_k(a,b)\)) in \(T\). In this paper, we prove that a connected local tournament \(T\) is completely strong path-connected if and only if for each arc \((a,b) \in A\), there exist \(P_2(a,b)\) and \(P’ _2(a,b)\) in \(T\), and \(T\) is not of \(T_1 \ncong T_0\)-\(D’_8\)-type digraph and \(D_8\).

It was proved by Ellingham \((1984)\) that every permutation graph either contains a subdivision of the Petersen graph or is edge-\(3\)-colorable. This theorem is an important partial result of Tutte’s Edge-\(3\)-Coloring Conjecture and is also very useful in the study of the Cycle Double Cover Conjecture. The main result in this paper is that every permutation graph contains either a subdivision of the Petersen graph or two \(4\)-circuits and therefore provides an alternative proof of the theorem of Ellingham. A corollary of the main result in this paper is that every uniquely edge-\(3\)-colorable permutation graph of order at least eight must contain a subdivision of the Petersen graph.

In this paper, the \(k\)-exponent and the \(k\)th upper multiexponent of primitive nearly reducible matrices are obtained and a bound on the \(k\)th lower multiexponent of this kind of matrices is given.

We call a simple \(t-(v,k)\) trade with maximum volume a maximal trade. In this paper, except for \(v = 6m+5\), \(m \geq 3\), maximal \(2-(v, 3)\) trades for all \(v\)’s are determined. In the latter case a bound for the volume of these trades is given.

Balanced ternary and generalized balanced ternary designs are constructed from any \((v, b, r, k)\) designs. These results generalise the earlier results of Diane Donovan ( 1985 ).