Let \(G\) be a graph of order \(n\), and let \(n = \sum_{i=1}^{k}a^i\) be a partition of \(n\) with \(a_i \geq 2\). Let \(v_1, \ldots, v_k\) be given distinct vertices of \(G\). Suppose that the minimum degree of \(G\) is at least \(3k\). In this paper, we prove that there exists a decomposition of the vertex set \(V(G) = \bigcup_{i=1}^k A_i\) such that \(|A_i| = a_i\), \(v_i \in A_i\), and the subgraph induced by \(A_i\) contains no isolated vertices for all \(i, 1 \leq i \leq k\).

Let \(G\) be a graph of order \(n \geq 4k\) and let \(S\) be the graph obtained from \(K_4\) by removing two edges which have a common vertex. In this paper, we prove the following theorem:
A graph \(G\) of order \(n \geq 4k\) with \(\sigma_2(G) \geq n+k\) has \(k\) vertex-disjoint \(S\).This theorem implies that a graph \(G\) of order \(n = 4k\) with \(\sigma_2(G) \geq 5k\) has an \(S\)-factor.

The reconstruction number \(rn(G)\) of graph \(G\) is the minimum number of vertex-deleted subgraphs of \(G\) required in order to identify \(G\) up to isomorphism. Myrvold and Molina have shown that if \(G\) is disconnected and not all components are isomorphic then \(rn(G) = 3\), whereas, if all components are isomorphic and have \(c\) vertices each, then \(rn(G)\) can be as large as \(c + 2\). In this paper we propose and initiate the study of the gap between \(rn(G) = 3\) and \(rn(G) = c + 2\). Myrvold showed that if \(G\) consists of \(p\) copies of \(K_c\), then\(rn(G) = c + 2\). We show that, in fact, this is the only class of disconnected graphs with this value of \(rn(G)\). We also show that if \(rn(G) \geq c + 1\) (where \(c\) is still the number of vertices in any component), then, again, \(G\) can only be copies of \(K_c\). It then follows that there exist no disconnected graphs \(G\) with \(c\) vertices in each component and \(rn(G) = c + 1\). This poses the problem of obtaining for a given \(c\), the largest value of \(t = t(c)\) such that there exists a disconnected graph with all components of order \(c\), isomorphic and not equal to \(K_c\), and is such that \(rn(G) = t\).

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].