Let \(T\) be a partially ordered set whose Hasse diagram is a binary tree and let \(T\) possess a unique maximal element \(1_T\). For a natural number \(n\), we compare the number \(A_T^n\) of those chains of length \(n\) in \(T\) that contain \(1_T\) and the number \(B_T^n\) of those chains that do not contain \(1_T\). We show that if the depth of \(T\) is greater or equal to \(2n + [ n \log n ]\) then \(B_T^n > A_T^n\).

The boundedness and compactness of the weighted composition operator from logarithmic Bloch spaces to a class of weighted-type spaces are studied in this paper.

S.M. Lee proposed the conjecture: for any \(n > 1\) and any permutation \(f\) in \(S(n)\), the permutation graph \(P(P_n, f)\) is graceful. For any integer \(n > 1\), we discuss gracefulness of the permutation graphs \(P(P_n, f)\) when \(f = (123), (n-2, n-1, n), (i, i+1), 1 \leq i \leq n-1, (12)(34)\ldots(2m-1, 2m), 1 \leq m \leq \frac{n}{2}\), and give some general results.

A double-loop network (DLN) \(G(N;r,s)\) is a digraph with the vertex set \(V = \{0,1,\ldots, N-1\}\) and the edge set \(E=\{v \to v+r \pmod{N} \text{ and } v \to v+s \pmod{N} | v \in V\}\). Let \(D(N;r,s)\) be the diameter of \(G(N;r,s)\) and let us define \(D(N) = \min\{D(N;r,s) | 1 \leq r < s < N \text{ and } \gcd(N,r,s) = 1\}\), \(D_1(N) = \min\{D(N;1,s) | 1 < s 0\)). Coppersmith proved that there exists an infinite family of \(N\) for which the minimum diameter \(D(N) \geq \sqrt{3N} + c(\log N)^{\frac{1}{4}}\), where \(c\) is a constant.

In this paper, we consider cycle-partition problems which deal with the case when both vertices and edges are specified and we require that they should belong to different cycles. Minimum degree and degree sum conditions are given, which are best possible.

In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices.

We correct and improve results from a recent paper by G. Ren and U. Kahler, which characterizes the Bloch, the little Bloch and Besov space of harmonic functions on the unit ball \({B} \subset \mathbb{R}^n\).

Let \(G\) be a \(2\)-connected graph with maximum degree \(\Delta(G) \geq d\), and let \(x\) and \(z\) be distinct vertices of \(G\). Let \(W\) be a subset of \(V(G) \setminus \{x, z\}\) such that \(|W| \leq d – 1\). Hirohata proved that if \(\max\{d_G(u), d_G(v)\} \geq d\) for every pair of vertices \(u, v \in V(G) \setminus \{x, z\} \cup W\) such that \(d_G(u, v) = 2\), then \(x\) and \(z\) are joined by a path of length at least \(d – |W|\). In this paper, we show that if \(G\) satisfies the conditions of Hirohata’s theorem, then for any given vertex \(y\) such that \(d_G(y) \geq d\), \(x\) and \(z\) are joined by a path of length at least \(d – |W|\) which contains \(y\).

For any positive integer \(k\), there exists a smallest positive integer \(N\), depending on \(k\), such that every \(2\)-coloring of \(1, 2, \ldots, N\) contains a monochromatic solution of the equation \(x + y + kz = 3w\). Based on computer checks, Robertson and Myers in \([5]\) conjectured values for \(N\) depending on the congruence class of \(k\) (mod \(9\)). In this note, we establish the values of \(N\) and find that in some cases they depend on the congruence class of \(k\) (mod \(27\)).

The support of a matrix \(M\) is the \((0, 1)\)-matrix with \(ij\)-th entry equal to \(1\) if the \(ij\)-th entry of \(M\) is non-zero, and equal to \(0\), otherwise. The digraph whose adjacency matrix is the support of \(M\) is said to be the digraph of \(M\). In this paper, we observe some general properties of digraphs of unitary matrices.