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 ).

A graph is called \(K_{1,r}\)-free if it does not contain \(K_{1,r}\) as an induced subgraph. In this paper we generalize a theorem of Markus for Hamiltonicity of \(2\)-connected \(K_{1,r}\)-free (\(r \geq 5\)) graphs and present a sufficient condition for \(1\)-tough \(K_{1,r}\)-free (\(r \geq 4\)) graphs to be Hamiltonian.

Minimum degree two implies the existence of a cycle. Minimum degree \(3\) implies the existence of a cycle with a chord. We investigate minimum degree conditions to force the existence of a cycle with \(k\) chords.

Let \(T = (V, E)\) be a tree on \(|V| = n\) vertices. \(T\) is graceful if there exists a bijection \(f : V \to \{0,1,\dots, n-1\}\) such that \(\{|f(u) – f(v)| \mid uv \in E\} = \{1,2,\dots,n-1\}\). If, moreover, \(T\) contains a perfect matching \(M\) and \(f\) can be chosen in such a way that \(f(u) + f(v) = n-1\) for every edge \(uv \in M\) (implying that \(\{|f(u) – f(v)| \mid uv \in M\} = \{1,3,\dots,n-1\}\)), then \(T\) is called strongly graceful. We show that the well-known conjecture that all trees are graceful is equivalent to the conjecture that all trees containing a perfect matching are strongly graceful. We also give some applications of this result.