Some results on combinatorial aspects of block designs using the complementary property have been obtained. The results pertain to non-existence of partially balanced incomplete block (PBIB) designs and identification of new \(2\)-associate and \(3\)-associate PBIB designs. A method of construction of extended group divisible (EGD) designs with three factors using self-complementary rectangular designs has also been given. Some rectangular designs have also been obtained using self-complementary balanced incomplete block designs. Catalogues of EGD designs and rectangular designs obtainable from these methods of construction, with number of replications \(\leq 10\) and block size \(\leq 10\) have been prepared.

For any simple graph \(H\), let \(\sigma(H, n)\) be the minimum \(m\) so that for any realizable degree sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with sum of degrees at least \(m\), there exists an \(n\)-vertex graph \(G\) witnessing \(\pi\) that contains \(H\) as a weak subgraph. Let \(F_{k}\) denote the friendship graph on \(2k+1\) vertices, that is, the graph of \(k\) triangles intersecting in a single vertex. In this paper, for \(n\) sufficiently large, \(\sigma(F_{k},n)\) is determined precisely.

Let \(C\) be a plane convex body, and let \(l(ab)\) be the Euclidean length of a longest chord of \(C\) parallel to the segment \(ab\) in \(C\). By the relative length of \(ab\) in a convex body \(C\), we mean the ratio of the Euclidean length of \(ab\) to \(\frac{l(ab)}{2}\). We say that a side \(ab\) of a convex \(n\)-gon is relatively short if the relative length of \(ab\) is not greater than the relative length of a side of the regular \(n\)-gon. In this article, we provide a significant sufficient condition for a convex hexagon to have a relatively short side.

This paper studies families of self-orthogonal codes over \(\mathbb{Z}_4\). We show that the simplex codes (of Type \(a\) and Type \(\beta\)) are self-orthogonal. We answer the question of \(\mathbb{Z}_4\)-linearity for some codes obtained from projective planes of even order. A new family of self-orthogonal codes over \(\mathbb{Z}_4\) is constructed via projective planes of odd order. Properties such as self-orthogonality, weight distribution, etc. are studied. Finally, some self-orthogonal codes constructed from twistulant matrices are presented.

A complete paired comparison digraph \(D\) is a directed graph in which \(xy\) is an arc for all vertices \(x,y\) in \(D\), and to each arc we assign a real number \(0 \leq a \leq 1\) called a weight such that if \(xy\) has weight \(a\) then \(yx\) has weight \(1 – a\). We say that two vertices \(x, y\) dominate a third \(z\) if the weights on \(xz\) and \(yz\) sum to at least \(1\). If \(x\) and \(y\) dominate all other vertices in a complete paired comparison digraph, then we say they are a dominant pair. We construct the domination graph of a complete paired comparison digraph \(D\) on the same vertices as \(D\) with an edge between \(x\) and \(y\) if \(x\) and \(y\) form a dominant pair in \(D\). In this paper, we characterize connected domination graphs of complete paired comparison digraphs. We also characterize the domination graphs of complete paired comparison digraphs with no arc weight of \(.5\).

A graph \(G\) is a \((d,d+k)\)-graph, if the degree of each vertex of \(G\) is between \(d\) and \(d+k\). Let \(p > 0\) and \(d+k \geq 2\) be integers. If \(G\) is a \((d,d+k)\)-graph of order \(n\) with at most \(p\) odd components and without a matching \(M\) of size \(2|M| = n – p\), then we show in this paper that

1. \(n \geq 2d+p+2\) when \(p \leq k-2\),
2. \(n \geq 2\left\lceil \frac{d(p+2)}{k} \right\rceil +p +2\) when \(p \geq k-1\).

Corresponding results for \(0 \leq p \leq 1\) and \(0 \leq k \leq 1\) were given by Wallis \([6]\), Zhao \([8]\), and Volkmann \([5]\).

Examples will show that the given bounds (i) and (ii) are best possible.

In this paper, we prove that the cycle \(C_n\) with parallel chords and the cycle \(C_n\) with parallel \(P_k\)-chords are cordial for any odd positive integer \(k \geq 3\) and for all \(n \geq 4\) except for \(n \equiv 4r + 2, r \geq 1\). Further, we show that every even-multiple subdivision of any graph \(G\) is cordial and we show that every graph is a subgraph of a cordial graph.

A hypergraph is linear if no two distinct edges intersect in more than one vertex. A long standing conjecture of Erdős, Faber, and Lovász states that if a linear hypergraph has \(n\) edges, each of size \(n\), then its vertices can be properly colored with \(n\) colors. We prove the correctness of the conjecture for a new, infinite class of linear hypergraphs.

We use a computer to show that the crossing number of generalized Petersen graph \(P(10,3)\) is six.

Let \(G\) be a graph in which each vertex has been coloured using one of \(k\) colours, say \(c_1,c_2,\ldots,c_k\) If an \(m\)-cycle \(C\) in \(G\) has \(n_i\) vertices coloured \(c_i\), \(i = 1,2,\ldots,k\), and \(|n_i – n_j| \leq 1\) for any \(i,j \in \{1,2,\ldots,k\}\), then \(C\) is equitably \(k\)-coloured. An \(m\)-cycle decomposition \(C\) of a graph \(G\) is equitably \(k\)-colourable if the vertices of \(G\) can be coloured so that every \(m\)-cycle in \(C\) is equitably \(k\)-coloured. For \(m = 4,5\) and \(6\), we completely settle the existence problem for equitably \(2\)-colourable \(m\)-cycle decompositions of complete graphs and complete graphs with the edges of a \(1\)-factor removed.