We take a special \(1\)-factorization of \(K_{n,n}\), and investigate the subgraphs suborthogonal to the \(1\)-factorization. Some interesting results are obtained, including an identity involving \(n^n\) and \(n!\) and a property of permutations.

An extended Mendelsohn triple system of order \(v\) (EMTS(\(v\))) is a collection of cyclically ordered triples of the type \([x,y,z], [x,x,y]\), or \([x,x,x]\) chosen from a \(v\)-set, such that each ordered pair (not necessarily distinct) belongs to exactly one triple. If such a design with parameters \(v\) and \(a\) exist, then they will have \(b_{v,a}\) blocks, where \(b_{v,a} = (v^2 + 2a)/3\). In this paper, we show that there are two (not necessarily distinct) EMTS(\(v\))’s with common triples in the following sets:

\(\{0,1,2,\ldots,b_v-4,b_v-2,b_v\}\), if \(v \neq 6\); and

\(\{0,1,2,\ldots,b_v-4,b_v-2\}\), if \(v = 6\),

where \(b_v\) is \(b_{v,v-1}\) if \(v \equiv 2 \pmod{3}\); \(b_{v,v}\) if \(v \not\equiv 2 \pmod{3}\).

Dudeney’s round table problem was proposed about one hundred years ago. It is already solved when the number of people is even, but it is still unsettled except for only a few cases when the number of people is odd.

In this paper, a solution of Dudeney’s round table problem is given when \(n = p+2\), where \(p\) is an odd prime number such that \(2\) is the square of a primitive root of \(\mathrm{GF}(p)\), and \(p \equiv 3 \pmod{4}\).

The number \(g^{(4)}_{2}\) is the minimal number of blocks that contain all pairs from a set of \(8\) elements exactly twice under the restriction that the longest block has size \(4\) (this longest block need not be unique). Thus the blocks have lengths \(2, 3\), and \(4\). We show that there are three solutions to this problem.

The \(n \times n\) primitive nearly reducible Boolean matrices whose \(k\)-exponents (\(1 \leq k \leq n\)) achieve the maximum value are characterized.

A graph is said to be \(k\)-covered if for each edge \(xy\), \(deg(x) = k\) or \(deg(y) = k\). In this paper, we characterize the \(3\)-covered quadrangulations of closed surfaces.

A graceful graph with \(n\) edges and \(n+1\) vertices is called a vertex-saturated graph. Each graceful graph corresponds to a vertex-saturated graph. Four classes of graceful graphs associated with vertex-saturated graphs are presented. Three of which generalize the results of [1], [2] and [5].

We correct an earlier theorem and reprove its consequences regarding \(c\)-BRDs with \(v \equiv 5, 8 \pmod{12}\). The original conclusions remain valid.

The type of a vertex \(v\) in a \(p\)-page book-embedding is the \(p \times 2\) matrix of nonnegative integers

\[{r}(v) =
\left(
\begin{array}{ccccc}
l_{v,1} & r_{v,1} \\
. & . \\
. & . \\
. & . \\
l_{v,p} & r_{v,p} \\
\end{array}
\right),\]

where \(l_{v,i}\) (respectively, \(r_{v,i}\)) is the number of edges incident to \(v\) that connect on page \(i\) to vertices lying to the left (respectively, to the right) of \(v\). The type number of a graph \(G\), \(T(G)\), is the minimum number of different types among all the book-embeddings of \(G\). In this paper, we disprove the conjecture by J. Buss et al. which says for \(n \geq 4\), \(T(L_n)\) is not less than \(5\) and prove that \(T(L_n) = 4\) for \(n \geq 3\).