The convex polyhedron of all real-valued monotone functions defined on a finite poset is an unbounded variant of the order polytope described by Stanley. If the undirected covering graph of the poset is acyclic, then the lattice of non-empty faces of this polyhedron is a Boolean lattice. In every other case, both semimodularity and dual semimodularity fail.

In a paper of Cockayne et al., the authors establish an upper and a lower bound for the dominating number of the complete grid graph \(G_{n,n}\), of order \(n^2\). Namely, they proved a “formula”, and cited two questions of Paul Erdős. One of these questions was “Can we improve the order of the difference between lower and upper bounds from \(\frac{n}{5}\) to \(\frac{n}{2}\)?”. Our aim here is to give a positive answer to this question.

Let \(D = (V_1, V_2; A)\) be a directed bipartite graph with \(|V_1| = |V_2| = n \geq 2\). Suppose that \(d_D(x) + d_D(y) \geq 3n\) for all \(x \in V_1\) and \(y \in V_2\). Then, with one exception, \(D\) contains two vertex-disjoint directed cycles of lengths \(2s\) and \(2t\), respectively, for any two positive integers \(s\) and \(t\) with \(s+t \leq n\).

The edge clique graph of a graph \(G\) is one having as vertices the edges of \(G\), two vertices being adjacent if the corresponding edges of \(G\) belong to a common clique.

Recently, Hsu and Shiue [10] obtained a kind of generalized Stirling number pairs with three free parameters and proved some of its properties. Here, some properties analogous to those of ordinary Stirling numbers are investigated, viz. horizontal recurrence relations, vertical recurrence relations, rational generating function, and explicit formulas. Furthermore, a kind of infinite sum which is useful in some combinatorial applications of the generalized Stirling numbers, is evaluated.

Clique graphs of several classes of graphs have been already characterized. Trees, interval graphs, chordal graphs, block graphs, clique-Helly graphs are some of them. However, no characterization of clique graphs of circular-arc graphs and some of their subclasses is known. In this paper, we present a characterization theorem of clique graphs of Helly circular-arc graphs and prove that this subclass of circular-arc graphs is properly contained in the intersection between proper circular-arc graphs, clique-Helly circular-arc graphs and Helly circular-arc graphs. Furthermore, we prove properties about the \(2^{\text{nd}}\) iterated clique graph of this family of graphs.

Let \(g: \mathbb{F}^m \to \mathbb{F}\) be a linear function on the vector space \(\mathbb{F}^m\) over a finite field \(\mathbb{F}\). A subset \(S \subsetneqq \mathbb{F}\) is called \(g\)-thin iff \(g(S^m) \subsetneqq \mathbb{F}\). In case \(\mathbb{F}\) is the field \(\mathbb{Z}_p\) of odd prime order, if \(S\) is \(g\)-thin and if \(m\) divides \(p-1\), then it is shown that \(|S| \leq \frac{p-1}{m}\). We also show that in certain cases \(S\) must be an arithmetic progression, and the form of the linear function \(g\) can be characterized.

A family \(\mathcal{F}\) of finite sets is said to have property \(B\) if there exists a set \(S\) such that \(0 < |{S} \cap F| < |F|\) for all \(F \in \mathcal{F}\). Denote by \(m_N(n)\) the least integer \(m\) for which there exists a family \(\mathcal{F}\) of \(m\) \(n\)-element subsets of a set \(V\) of size \(N\) such that \(\bigcup \mathcal{F} = V\) and which does not have property \(B\). We give constructions which yield upper bounds for \(m_N(4)\) for certain values of \(N\).

Let \(G\) be a connected graph and \(\mathcal{V}^*\) the set of all spanning trees except stars in \(G\). An edge in a spanning tree is called `inner’ if the edge is not incident to endvertices. Define an adjacency relation in \(\mathcal{V}^*\) as follows: two spanning trees \(t_1\) and \(t_2 \in \mathcal{V}^*\) are called to be adjacent if there exist inner edges \(e_i \in E(t_i)\) such that \(t_1 – e_1 = t_2 – e_2\). The resultant graph is a subgraph of the tree graph, and we call it simply a trunk graph. The purpose of this paper is to show that if a \(2\)-connected graph with at least five vertices is \(k\)-edge connected, then its trunk graph is \((k-1)\)-connected.

Let \(\tau(n)\) denote Ramanujan’s tau function. We obtain an identity that involves \(\tau(n)\) and \(\sigma(n)\), as well as some apparently new congruence properties of \(\tau(n)\) with respect to the moduli \(23\) and \(5\).