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 \(5\).

For loopless multigraphs \(G\), the total choice number is asymptotically equal to its fractional counterpart as the latter invariant tends to infinity. If \(G\) is embedded in the plane, then the edge-face and entire choice numbers exhibit the same “asymptotically good” behaviour. These results are based mainly on an analogous theorem of Kahn [5] for the list-chromatic index. Together with work of Kahn and others, our three results give a complete answer to a natural question: which of the seven invariants associated with list-colouring the nonempty subsets of \(\{V, E, F\}\) are asymptotically good?

In 1970, Behzad, Chartrand and Wall conjectured that the girth of every \(r\)-regular digraph \(G\) of order \(n\) is at most \(\left\lceil \frac{n}{r} \right\rceil\). The conjecture follows from a theorem of Menger and Dirac if \(G\) has strong connectivity \(x = r\). We show that any digraph with minimum in-degree and out-degree at least \(r\) has girth at most \(\left\lceil \frac{n}{r} \right\rceil\) if \(\kappa = r – 1\). We also find from the literature a family of counterexamples to a conjecture of Seymour.

In this paper, we give an alternative proof for the fact that the graph obtained by overlapping the cycle \(C_m\) (\(m \geq 3\)) and the complete bipartite graph \(K_{2,s}\) (\(s \geq 1\)) at an edge is uniquely determined by its chromatic polynomial. This result provides a partial solution to a question raised in [7].

Let \(G\) be a simple graph with \(n\) vertices. \(p(G,k)\) denotes the number of ways in which one can select \(k\) independent edges in \(G\) (\(k \geq 1\)). Let \(p(G,0) = 1\) for all \(G\).
The matching polynomial \(\alpha(G)\) of a graph \(G\) is given by:
\[\alpha(G) = \alpha(G,x) = \sum_{k=0}^{\left[\frac{n}{2}\right]} (-1)^k p(G,k) x^{n-2k}\]

In this article, we give the matching polynomials of the complete \(n\)-partite graph with a differential operator.