A known theorem of Bigalke and Jung says that the only nonhamiltonian, tough graph \(G\) with \(\alpha(G) \leq H(G) + 1\), where \(H(G) \geq 3\), is the Petersen graph. In this paper we characterize all nonhamiltonian, tough graphs having k total vertex (i.e. adjacent to all others) with \(\alpha(G) \leq k+ 2\) (Theorem 3).

Given a sequence \(S: d_1, d_2, \ldots, d_p\) of non-negative integers, we give necessary and sufficient conditions for a subsequence of \(S\) with \(p – 1\) terms to be graphical.

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 \([3]\).