The upper chromatic number \(\overline{\chi}_u(\mathcal{H})\) of a \(C\)-hypergraph \(\mathcal{H} = (X, C)\) is the maximum number of colors that can be assigned to the vertices of \(\mathcal{H}\) in such a way that each \(C \in \mathcal{C}\) contains at least a monochromatic pair of vertices. This paper gives an upper bound for the upper chromatic number of Steiner triple systems of order \(n\) and proves that it is best possible for any \(n (\equiv 1 \text{ or } 3 \pmod{6})\).

A map is called Unicursal if it has exactly two vertices of odd valency. A near-triangulation is a map with all but one of its faces triangles. We use the enufunction approach to enumerate rooted Unicursal planar near-triangulations with the valency of the root-face and the number of non-rooted faces as parameters.

An edge coloring is proper if no two adjacent edges are assigned the same color and vertex-distinguishing proper coloring if it is proper and incident edge sets of every two distinct vertices are assigned different sets of colors. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph \(G\) is denoted by \(\overline{\chi}'(G)\). In this paper, we prove that \(\overline{\chi}'(G) \leq \Delta(G) + {4}\) if \(G = (V, E)\) is a connected graph of order \(n \geq 3\) and \(\sigma_2(G) \geq n\), where \(\sigma_2(G) = \min\{d(x) + d(y) | xy \in E(G)\}\).

In this paper, we study the minimum distance between the set of bent functions and the set of \(1\)-resilient Boolean functions and present lower bounds on that. The first bound is proved to be tight for functions up to \(10\) input variables and a revised bound is proved to be tight for functions up to \(14\) variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of \(1\)-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied up to \(14\)-variable functions and we show that the construction provides a large class of \(1\)-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of \(8\)-variable, \(1\)-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from \(10\) variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.

In this paper, we characterize the variety of quasi-groups isotopic to abelian groups by four-variable identities.

A direct method for constructing large sets of \(t\)-designs is based on the concept of assembling orbits of a permutation group \(G\) on \(k\)-subsets of a \(v\)-set into block sets of \(t\)-designs so that these designs form a large set. If \(G\) is \(t\)-homogeneous, then any orbit is a \(t\)-design and therefore we obtain a large set by partitioning the set of orbits into parts consisting of the same number of \(k\)-subsets. In general, it is hard to find such partitions. We solve this problem when orbit sizes are limited to two values. We then use its corollaries to obtain some results in a special case in which a simple divisibility condition holds and no knowledge about orbit sizes is assumed.

Dean \(([3])\) shows that if \(G\) be a \(k\)-connected graph such that any fragment whose neighborhood contains an edge has cardinality exceeding \(\frac{k}{2}\), then the subgraph \(H = (V(G), E_k(G))\) formed by \(V(G)\) and the \(k\)-contractible edges of \(G\) is \(2\)-connected. In this paper, we show that for \(k = 4\), Dean’s result holds when reduced \(\frac{k}{2}\) to \(\frac{k}{4}\). But for \(k \geq 5\), we give a counterexample to show that it is false and give a lower bound of the number of \(k\)-contractible edges for \(k = 5\).

Let \(G\) be a finite simple \(\chi\)-chromatic graph and \(\mathcal{L} = \{L_u\}_{u\in V(G)}\) be a list assignment to its vertices with \(L_u \subseteq \{1,…,\chi\}\). A list colouring problem \((G, \mathcal{L})\) with a unique solution for which the sum \(\sum_{u\in V(G)}|L_u|\) is maximized, is called a \(maximum\; \chi-list \;assignment\) of \(G\). In this paper, we prove a \(Circuit\; Simulation\) Lemma that, strictly speaking, makes it possible to simulate any Boolean function by \(effective\) 3-colourings of a graph that is \(polynomial-time \;constructable\) from the Boolean function itself. We use the lemma to simply prove some old results as corollaries, and also we prove that the following decision problem, related to the computation of the fixing number of a graph [Daneshgar \(1997\), Daneshgar and Naserasr, Ars Combin. \(69\) \((2003)\)], is \(\sum_{2}^{P}\)-complete.

In this paper, we give another proof for labeled spanning forests of the complete bipartite graph \(K_{m,n}\), and obtain two Abel-type polynomials. And then we investigate the enumeration of non-trivial rooted labeled spanning forests of the complete bipartite graph \(K_{m,n}\).

In this paper, we investigate the global behavior of the difference equation

\[x_{n+1} = \frac{\alpha x_{n-k}}{\beta+\gamma x_{n-(k+1)}^p},\text{n=1,2,}\ldots\]

with non-negative parameters and non-negative initial conditions, where \(k\) is an odd number.