Let \(G\) be a graph and let \(\sigma_k(G)\) be the minimum degree sum of an independent set of \(k\) vertices. For \(S \subseteq V(G)\) with \(|S| \geq k\), let \(\Delta_k(S)\) denote the maximum value among the degree sums of the subset of \(k\) vertices in \(S\). A cycle \(C\) of a graph \(G\) is said to be a dominating cycle if \(V(G \setminus C)\) is an independent set. In \([2]\), Bondy showed that if \(G\) is a \(2\)-connected graph with \(\sigma_3(G) \geq |V(G)| + 2\), then any longest cycle of \(G\) is a dominating cycle. In this paper, we improve it as follows: if \(G\) is a 2-connected graph with \(\Delta_3(S) \geq |V(G)| + 2\) for every independent set \(S\) of order \(\kappa(G) + 1\), then any longest cycle of \(G\) is a dominating cycle.

Let \(B\) be an \(m \times n\) array in which each symbol appears at most \(k\) times. We show that if \(k \leq \frac{n(n-1)}{8(m+n-2)} + 1\) then \(B\) has a transversal.

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 \({B} \subset \mathbb{R}^n\).