An \(f\)-coloring of a graph \(G\) is a coloring of edges of \(E(G)\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\) and denoted by \(\chi’_f(G)\). Any simple graph \(G\) has the \(f\)-chromatic index equal to \(\Delta_f(G)\) or \(\Delta_f(G) + 1\), where \(\Delta_f(G) = \max_{v \in V}\{\lceil \frac{d(v)} {f(v)}\rceil\}\). If \(\chi’_f(G) = \Delta_f(G)\), then \(G\) is of \(C_f\) \(1\); otherwise \(G\) is of \(C_f\) \(2\). In this paper, two sufficient conditions for a regular graph to be of \(C_f\) \(1\) or \(C_f\) \(2\) are obtained and two necessary and sufficient conditions for a regular graph to be of \(C_f\) \(1\) are also presented.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f: V(G) \to A\) induces an edge labeling \(f^*: E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \text{card}\{v \in V(G): f(v) = i\}\) and \(e_f(i) = \text{card}\{e \in E(G): f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i,j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A-friendly\) if \(|v_f(i) – v_f(j)| \leq 1\) for all \((i,j) \in A \times A\). If \(c(f)\) is a \((0,1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the \({friendly index set}\) of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)|: \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In this paper, we determine the friendly index set of cycles, complete graphs, and some bipartite graphs.

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.