We establish some basic facts about sign-patterns of orthogonal matrices, and use these facts to characterize the sign-nonsingular matrices which are sign-patterns of orthogonal matrices.

In this paper, we give some properties of balanced labeling, prove that the graph \((m^2 + 1)C_4\) is balanced, and also solve the balanceness of snakes \(C_m(n)\).

In this note, we verify two conjectures of Catlin in [J.Graph Theory \(13 (1989)465-483\)] for graphs with at most \(11\) vertices. These are used to prove the following theorem, which improves prior results in \([10]\) and \([13]\):

Let \(G\) be a 3-edge-connected simple graph with order \(n\). If \(n\) is large and if for every edge \(uv \in E(G)\), \(d(u) + d(v) \geq \frac{n}{6} – 2\), then either \(G\) has a spanning eulerian subgraph or $G$ can be contracted to the Petersen graph.

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.