Let \(D\) be a strictly disconnected digraph with \(n\) vertices. A common out-neighbor (resp. in-neighbor) of a pair of vertices \(u\) and \(v\) is a vertex \(x\) such that \(ux\) and \(vx\) (resp. \(xu\) and \(xv\)) are arcs of \(D\). It is shown that if

\[d^+(u_1) + d^+(v_1) + d^-(u_2) + d^-(v_2) > 2n-1\]

for any pair \(u_1, v_1\) of nonadjacent vertices with a common out-neighbor and any pair \(u_2, v_2\) of nonadjacent vertices with a common in-neighbor, then \(D\) contains a directed Hamiltonian cycle.

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.