For each integer \(m \geq 1\), consider the graph \(G_m\) whose vertex set is the set \(\mathbb{N} = \{0,1,2,\ldots\}\) of natural numbers and whose edges are the pairs \(xy\) with \(y = x+m\), \(y = x-m\), \(y = mx\), or \(y = \frac{x}{m}\). Our aim in this note is to show that, for each \(m\), the graph \(G_m\) contains a Hamilton path. This answers a question of Lichiardopol.

Given a partial \(K_4\)-design \((X, {P})\), if \(x \in X\) is a vertex which occurs in exactly one block of \({P}\), then call \(x\) a free vertex. In this paper, a technique is described for obtaining a cubic embedding of any partial \(K_4\)-design with the property that every block in the partial design contains at least two free vertices.

The average distance \(\mu(D)\) of a strong digraph \(D\) is the average of the distances between all ordered pairs of distinct vertices of \(D\). Plesnik \([6]\) proved that if \(D\) is a strong tournament of order \(n\), then \(\mu(D) \leq \frac{n+4}{6} + \frac{1}{n}\). In this paper, we show that if \(D\) is a \(k\)-connected tournament of order \(n\), then \(\mu(D) \leq \frac{n}{6k} + \frac{19}{6} + \frac{k}{n}\). We demonstrate that, apart from an additive constant, this bound is best possible.

A subset \(U\) of a set \(S\) with a binary operation is called avoidable if \(S\) can be partitioned into two subsets \(A\) and \(B\) such that no element of \(U\) can be written as a product of two distinct elements of \(A\) or as the product of two distinct elements of \(B\). The avoidable sets of the bicyclic inverse semigroup are classified.

Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by

\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]

Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by

\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]

We call the matrix \((a_{n,m})_{n,m\geq 0}\) an generalized Seidel matrix with a parameter pair \((\alpha, \beta)\). If \(\alpha = \beta = 1\), then this matrix is the classical Seidel matrix. For various different parameter pairs \((\alpha, \beta)\) we will impose some evenness or oddness conditions on the exponential generating functions of the initial sequence \(a_{0,m}\) and the final sequence \(a_{n,0}\) of a generalized Seidel matrix (i.e., we require that these generating functions or certain related functions are even or odd). These conditions imply that the initial sequences and final sequences are equal to well-known classical sequences such as those of the Euler numbers, the Genocchi numbers, and the Springer numbers.

As applications, we give a straightforward proof of the continued fraction representations of the ordinary generating functions of the sequence of Genocchi numbers. And we also get the continued fractions representations of the ordinary generating functions of the Genocchi polynomials, Bernoulli polynomials, and Euler polynomials. Lastly, we give some applications of congruences for the Euler polynomials.

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(f: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(\overline{f}(uv) = |f(u) – f(v)|\). Let \(v_f(0), v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0), e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – v_f(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).

In this paper, we give necessary and sufficient conditions for the cordiality of the \(t\)-ply \(P_t(u,v)\), i.e. a thread of ply number \(t\).

A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over \(\mathbb{F}_2\). Later, Bannai and Ozeki showed how to construct Jacobi forms with various index using a Jacobi polynomial corresponding to the binary codes. It generalizes Broué-Enguehard map. In this paper, we study Jacobi polynomial which corresponds to the codes over \(\mathbb{F}_{2f}\). We show how to construct Jacobi forms with various index over the totally real field. This is one of extension of Broué-Enguehard map.

The paper contains two main results. First, we obtain the chromatic polynomial on the \(n \times m\) section of the square lattice, solving a problem proposed by Read and Tutte \([5]\), the chromatic polynomial of the bracelet square lattice, and we find a recurrent-constructive process for the matrices of the \(k\)-colourings. The key concept for obtaining the inductive method is the compatible matrix.

Our second main result deals with the compatible matrix as the adjacency matrix of a graph. This represents a family of graphs, which is described.

Let \(G = (V, E)\) be a simple graph. For any real valued function \(f: V \to \mathbb{R}\), the weight of \(f\) is defined as \(f(V) = \sum f(v)\), over all vertices \(v \in V\). For positive integer \(k\), a total \(k\)-subdominating function (TkSF) is a function \(f: V \to \{-1,1\}\) such that \(f(N(v)) \geq k\) for at least \(k\) vertices \(v\) of \(G\). The total \(k\)-subdomination number \(\gamma^t_{ks}(G)\) of a graph \(G\) equals the minimum weight of a TKSF on \(G\). In the special case where \(k = |V|\), \(\gamma^t_{ks}(G)\) is the signed total domination number \([5]\). We research total \(k\)-subdomination numbers of some graphs and obtain a few lower bounds of \(\gamma^t_{ks}(G)\).

A convex hull of a set of points \(X\) is the minimal convex set containing \(X\). A box \(B\) is an interval \(B = \{x | x \in [a,b], a,b \in \mathbb{R}^n\}\). A box hull of a set of points \(X\) is defined to be the minimal box containing \(X\). Because both convex hulls and box hulls are closure operations of points, classical results for convex sets can naturally be extended for box hulls. We consider here the extensions of theorems by Carathéodory, Helly, and Radon to box hulls and obtain exact results.