Let \(G\) be a graph. A vertex subversion strategy of \(G\), \(S\), is a set of vertices in \(G\) whose closed neighborhood is deleted from \(G\). The survival-subgraph is denoted by \(G/S\). The vertex-neighbor-integrity of \(G\), \(VNI(G)\), is defined as:

\(VNI(G) = \min_{|S|} {|S| + w(G/S)}\)

where \(S\) is any vertex subversion strategy of \(G\), and \(w(G/S)\) is the maximum order of the components of \(G/S\). In this paper, we evaluate the vertex-neighbor-integrity of the powers of cycles, and show that among the powers of the \(n\)-cycle, the maximum vertex-neighbor-integrity is \(\left\lceil{2}\sqrt{n}\right\rceil – 3\) and the minimum vertex-neighbor-integrity is \(\left\lceil\frac{n}{2\left\lfloor\frac{n}{2}\right\rfloor} + 1\right\rceil\).

What is the 2-packing number of the \(1 \times m \times n\) complete grid graph? Fisher solved this for \(1 \times m \times n\) grids for all \(m\) and \(n\). We answer this for \(2 \times m \times n\) grids for all \(m\) and \(n\), and for \(3 \times 3 \times n\), \(3 \times 4 \times n\), \(3 \times 7 \times n\), \(4 \times 4 \times n\), and \(5 \times 5 \times n\) grids for all \(n\). Partial results are given for other sizes.

A Pandiagonal magic square (PMS) of order \(n\) is a square matrix which is an arrangement of integers \(0, 1, \ldots, n^2-1\) such that the sums of each row, each column, and each extended diagonal are the same. In this paper, we use the Step method to construct a PMS of order \(n\) for each \(n > 3\) and \(n\) is not singly-even. We discuss how to enumerate the number of PMSs of order \(n\) constructed by the Step method, and we get the number of nonequivalent PMSs of order \(8\) with the top left cell \(0\) is \(4,176,000\) and the number of nonequivalent PMSs of order \(9\) with the top left cell \(0\) is \(1,492,992\).

In this paper, we consider total clique covers and uniform intersection numbers on multifamilies. We determine the uniform intersection numbers of graphs in terms of total clique covers. From this result and some properties of intersection graphs on multifamilies, we determine the uniform intersection numbers of some families of graphs. We also deal with the \(NP\)-completeness of uniform intersection numbers.

An oriented triple system of order \(v\), denoted OTS\((v)\), is said to be \(d\)-cyclic if it admits an automorphism consisting of a single cycle of length \(d\) and \(v-d\) fixed points, \(d\geq 2\). In this paper, we give necessary and sufficient conditions for the existence of \(d\)-cyclic OTS\((v)\). We solve the analogous problem for directed triple systems.

Let \(A_m(n, k)\) denote the number of permutations of \(\{1, \ldots, n\}\) with exactly \(k\) rises of size at least \(m\). We show that, for each positive integer \(m\), the \(A_m(n, k)\) are asymptotically normal.

Let \(G\) be a graph of order \(n\) and \( X\) a given vertex subset of \(G\). Define the parameters:
\(\alpha(V) = \max\{|S| \mid S\}\) is an independent set of vertices of the subgraph \(G(X)\) in \(G\) induced by \(X\)
and
\(\sigma_k(X) = \min\{|\Sigma_{i=1}^{k}d(x_i)| \mid \{x_1,x_2,\ldots,x_k\} \}\) is an independent vertex set in \( G[X]\)
A cycle \(C\) of \(G\) is called \(X\)-longest if no cycle of \(G\) contains more vertices of \(X\) than \(C\). A cycle \(C’\) of \(G\) is called \(X\)-dominating if all neighbors of each vertex of \(X\setminus V(C)\) are on \(C\). In particular, \(G\) is \(X\)-eyclable if \(G\) has an \(X\)-cycle, i.e., a cycle containing all vertices of \(X\). Our main result is as follows:
If \(G\) is \(1\)-tough and \(\sigma_3(X) \geq n\), then \(G\) has an \(X\)-longest cycle \(C\) such that \(C\) is an \(X\)-dominating cycle and \(|V(C) \cap X| \geq \min\{|X|. |X| + \frac{1}{3}\sigma_3(X) – \sigma(X)\}\), which extends the well-known results of D. Bauer et al. [2] in terms of \(X\)-cyclability. Finally, if \(G\) is \(2\)-tough and \(\sigma_3(X) \geq n\), then \(G\) is \(X\)-cyelable.

In 1992, Mahmoodian and Soltankhah conjectured that, for all \(0 \leq i \leq t\), a \((v, k, t)\) trade of volume \(2^{t+1} – 2^{t-i}\) exists. In this paper we prove this conjecture and, as a corollary, show that if \(s \geq (2t – 1)2^t\) then there exists a \((v, k, t)\) trade of volume \(s\).

We prove two new characterization theorems for finite Moufang polygons, one purely combinatorial, another group-theoretical. Both follow from a result of Andries Brouwer on the connectedness of the geometry opposite a flag in a finite generalized polygon.

Cyclonomial coefficients are defined as a generalization of binomial coefficients. It is proved that each natural number can be expressed, in a unique way, as the sum of cyclonomial coefficients, satisfying certain conditions. This cyclonomial number system generalizes the well-known binomial number system. It appears that this system is the appropriate number system to index the words of the lexicographically ordered code \(L^q(n, k)\). This code consists of all words of length \(n\) over an alphabet of \(q\) symbols, such that the sum of the digits is constant. It provides efficient algorithms for the conversion of such a codeword to its index, and vice versa.