A series of partially balanced incomplete block design yields under certain
restrictions, a new series of BIB designs with parameters:
\[v=\binom{2s+1}{2}, b=\frac{1}{2}(s+1)\binom{2s+1}{s+1}\]
\[v=s \binom{2s-1}{s},k=s^2, \lambda=(s-1)\binom{2s-1}{s-1}\]
where \(s \geq 2\) is any positive integer.

A \(d\)-design is an \(n \times n\) \((0,1)\)-matrix \(A\) satisfying \(A^t A = \lambda J + {diag}(k_1 – \lambda, \ldots, k_n – \lambda)\), where \(A^t\) is the transpose of \(A\), \(J\) is the \(n \times n\) matrix of ones, \(k_j >\lambda > 0\) (\(1 \leq j \leq n\)), and not all \(k_i\)’s are equal. Ryser [4] and Woodall [6] showed that such an \(A\) has precisely two row sums \(r_1\) and \(r_2\) (\(r_1 > r_2\)) with \(r_1 + r_2 = n + 1\). Let \(e_1\) be the number of rows of \(A\) with sum \(r_1\). It is shown that if \(e_1 = 4\), then \(\lambda = 3\).

In this note we introduce a lemma which is useful in studying the chromaticity of graphs. As examples, we give a short proof for a conclusion in \([3]\).

The existence of difference sets in abelian \(2\)-groups is a recently settled problem \([5]\); this note extends the abelian constructs of difference sets to nonabelian groups of order \(64\).

We deal with conditions on the number of arcs sufficient for bipartite digraphs to have cycles and paths with specified properties.

The convex hull of graph \(G\), a notion born in the theory of random graphs, is the convex hull of the set in \(xy\)-plane obtained by representing each subgraph \(H\) of \(G\) by the point whose coordinates are the number of vertices and edges of \(H\).

In the paper, the maximum number of corners of the convex hull of an \(n\)-vertex graph, bipartite graph, and \(K({r})\)-free graph is found. The same question is posed for strictly balanced graphs.

Conjectured generalizations of Hadwiger’s Conjecture are discussed. Among other results, it is proved that if \(X\) is a set of \(1\), \(2\) or \(3\) vertices in a graph \(G\) that does not have \(K_6\) as a subcontraction, then \(G\) has an induced subgraph that is \(2\)-, \(3\)- or \(4\)-colourable, respectively, and contains \(X\) and at least a quarter, a third or a half, respectively, of the remaining vertices of \(G\). These fractions are best possible.

In 1967 Alspach [1] proved that every arc of a diregular tournament is contained in cycles of all possible lengths. In this paper, we extend this result to bipartite tournaments by showing that every arc of a diregular bipartite tournament is contained in cycles of all possible even lengths, unless it is isomorphic to one of the graphs \(F_{4k} \). Simultaneously, we also prove that an almost diregular bipartite tournament \(R\) is Hamiltonian if and only if \(|V_1| = |V_2|\) and \(R\) is not isomorphic to one of the graphs \(F_{4k-2}\), where \((V_1, V_2)\) is a bipartition of \(R\). Moreover, as a consequence of our first result, it follows that every diregular bipartite tournament of order \(p\) contains at least \(p/4\) distinct Hamiltonian cycles. The graphs \(F_r = (V, A)\), (\(r \geq 2\)) is a family of bipartite tournaments with \(V = \{v_1, v_2, \ldots, v_r\}\) and \(A = \{v_iv_j | j – i \equiv 1 \pmod{4}\}\).

In this paper we study the edge clique graph \(K(G)\) of many classes of intersection graphs \(G\) — such as graphs of boxicity \(\leq k\), chordal graphs and line graphs. We show that in each of these cases, the edge clique graph \(K(G)\) belongs to the same class as \(G\). Also, we show that if \(G\) is a \(W_4\)-free transitivity orientable graph, then \(K(G)\) is a weakly \( \theta \)-perfect graph.