In this paper, several constructions are presented for balanced incomplete block designs with nested rows and columns. Some of them refine theorems due to Hishida and Jimbo \([6]\) and Uddin and Morgan \([17]\), and some of them give parameters which have not been available before.

A vertex-distinguishing edge-coloring (VDEC) of a simple graph \(G\) which contains no more than one isolated vertex and no isolated edge is equitable (VDEEC) if the absolute value of the difference between the number of edges colored by color \(i\) and the number of edges colored by color \(j\) is at most one. The minimal number of colors needed such that \(G\) has a VDEEC is called the vertex-distinguishing equitable chromatic index of \(G\). In this paper, we propose two conjectures after investigating VDEECs on some special families of graphs, such as the stars, fans, wheels, complete graphs, complete bipartite graphs, etc.

The eccentricity \(e(v)\) of a vertex \(v\) in a strongly connected digraph \(G\) is the maximum distance from \(v\). The eccentricity sequence of a digraph is the list of eccentricities of its vertices given in non-decreasing order. A sequence of positive integers is a digraphical eccentric sequence if it is the eccentricity sequence of some digraph. A set of positive integers \(S\) is a digraphical eccentric set if there is a digraph \(G\) such that \(S = \{e(v), v \in V(G)\}\). In this paper, we present some necessary and sufficient conditions for a sequence \(S\) to be a digraphical eccentric sequence. In some particular cases, where either the minimum or the maximum value of \(S\) is fixed, a characterization is derived. We also characterize digraphical eccentric sets.

Let \(C_m\) be a cycle on \(m (\geq 3)\) vertices and let \(\ominus_{n-m}C_m\) denote the class of graphs obtained from \(C_m\) by adding \(n-m (\geq 1)\) distinct pendent edges to the vertices of \(C_m\). In this paper, it is proved that for every \(T\) in \(\ominus_{n-m}C_m\), the complete graph \(K_{2n+1}\) can be cyclically decomposed into the isomorphic copies of \(T\). Moreover, if \(m\) is even, then for every positive integer \(p\), the complete graph \(K_{2pn+1}\) can also be cyclically decomposed into the isomorphic copies of \(T\).

An aperiodic perfect map (APM) is an array with the property that each possible array of a given size, called a window, arises exactly once as a contiguous subarray in the array. In this paper, we give a construction method of an APM being a proper concatenation of some fragments of a given de Bruijn sequence. Firstly, we give a criterion to determine whether a designed sequence \(T\) with entries from the index set of a de Bruijn sequence can generate an APM. This implies a sufficient condition for being an APM. Secondly, two infinite families of APMs are given by constructions of corresponding sequences \(T\), respectively, satisfying the criterion.

We introduce a combinatorial shifting operation on multicomplexes that carries similar properties required for the ordinary shifting operation on simplicial complexes. A linearly colored simplicial complex is called shifted if its associated multicomplex is stable under defined operation. We show that the underlying simplicial subcomplex of a linearly shifted simplicial complex is shifted in the ordinary sense, while the ordinary and linear shiftings are not interrelated in general. Separately, we also prove that any linearly shifted complex must be shellable with respect to the order of its facets induced by the linear coloring. As an application, we provide a characterization of simple graphs whose independence complexes are linearly shifted. The class of graphs obtained constitutes a superclass of threshold graphs.

A local coloring of a graph \(G\) is a function \(c: V(G) \to \mathbb{N}\) having the property that for each set \(S \subseteq V(G)\) with \(2 \leq |S| \leq 3\), there exist vertices \(u,v \in S\) such that \(|c(u) – c(v)| \geq m_S\), where \(m_S\) is the size of the induced subgraph \(\langle S\rangle\). The maximum color assigned by a local coloring \(c\) to a vertex of \(G\) is called the value of \(c\) and is denoted by \(\chi_\ell(c)\). The local chromatic number of \(G\) is \(\chi_\ell(G) = \min\{\chi_\ell(c)\}\), where the minimum is taken over all local colorings \(c\) of \(G\). If \(\chi_\ell(c) = \chi_\ell(G)\), then \(c\) is called a minimum local coloring of \(G\). The local coloring of graphs introduced by Chartrand et al. in \(2003\). In this paper, following the study of this concept, first an upper bound for \(\chi_\ell(G)\) where \(G\) is not complete graphs \(K_4\) and \(K_5\), is provided in terms of maximum degree \(\Delta(G)\). Then the exact value of \(\chi_\ell(G)\) for some special graphs \(G\) such as the cartesian product of cycles, paths and complete graphs is determined.

Explicit expressions for all the primitive idempotents in the ring \(R_{2^n} = {F}_q[x]/(x^{2^n} – 1)\), where \(q\) is an odd prime power, are obtained. Some lower bounds on the minimum distances of the irreducible cyclic codes of length \(2^n\) over \({F}_q\) are also obtained.

In this note we prove that all connected Cayley graphs of every finite group \(Q \times H\) are \(1\)-factorizable, where \(Q\) is any non-trivial group of \(2\)-power order and \(H\) is any group of odd order.