Hall’s theorem on differences of bijections characterizes the multisets \(\{a_1,\ldots,a_{|G|}\}\) in a finite abelian group \(G\) that can be written in the form \( a_i=b_i-c_i, \) where both \(b_1,\ldots,b_{|G|}\) and \(c_1,\ldots,c_{|G|}\) are enumerations of \(G\). The necessary and sufficient condition is the zero-sum condition \( a_1+\cdots+a_{|G|}=0. \) This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset \(A\) of cardinality \(|G|\) can be represented as \( A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, \) where \(b\) and \(c\) are bijections onto \(G\). Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of \(A\) is trivial in \(G_{\rm ab}\). We show that this condition is not sufficient in general, even when the elements of \(A\) admit an ordering whose product is the identity in \(G\). The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of \(A\) into product-one words whose partial-product sets tile \(G\) by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in \(S_3\), and we extend the same obstruction to infinitely many finite nonabelian groups.

We investigate diagonal equations \(ax^{m}+by^{m}-cz^{m}=1\) over finite fields \(F\) using combinatorial designs naturally associated with \(F\). Building on prior work that resolved the case \(a=b=c=1\), we obtain exact formulas for the solutions when \(a=1\) and \(b=c\), under circularity assumptions. For general coefficients, we present an algorithm that determines whether a given instance can be reduced to the settled cases, or else identifies it as requiring brute-force computation.

In this paper, we expand our interest in the 16th Hilbert’s problem to acquire a comprehensive understanding of the maximum number of crossing limit cycles in \(\mathbb{R}^3\), specifically within a class of three- dimensional discontinuous piecewise differential system generated by two arbitrary Euler systems separated by the unit sphere \(\mathbb{S}^2=\{ (x,y,z) \in\mathbb{R}^3; x^2 + y^2 + z^2 = 1\}\).

Let \(G\) be a graph with no isolated vertices. A \(k\)-coupon coloring of \(G\) is an assignment of colors from \([k]=\{1,2,\ldots,k\}\) to the vertices of \(G\) such that the neighborhood of every vertex contains all colors from \([k]\). The maximum integer \(k\) for which a \(k\)-coupon coloring exists is called the coupon coloring number of \(G\), and is denoted by \(\chi_c(G)\). In this paper, we investigate coupon coloring in inflated graphs arising from various classes of graphs. In addition, we introduce new graph operations based on inflation and study their effect on the existence and behavior of coupon colorings. Our results contribute to a deeper understanding of how inflation based graph operations influence coupon coloring.

We recall the definition and properties of a moment sequence and show that all real sequences whose Hankel matrices have finite rank (see definition in the sequel) satisfy a homogeneous linear equation with constant coefficients. Then we analyze the cases in which a difference equation with constant coefficients and suitably chosen initial conditions and having as an input a positive moment sequence has a solution that is a positive moment sequence. We give one general simple result and give many examples illustrating the theory. The main result states that the roots of the odd multiplicity of the characteristic equation must lie outside the support of the measure that produces the moment sequence that is in the input and the initial conditions suitably chosen.

We present constructions of semi-magic squares of side \(n=2k\), whose entries are elements of a dihedral group \(D_{2k^2}\), for every \(n\equiv0\pmod4\).

A zero divisor graph on a finite commutative ring \(\mathfrak{R}\) is a graph with set of vertices consists of zero divisor elements \(Z(\mathfrak{R})\) of the ring, and we have an edge between any two elements in \(\mathfrak{R}\) if their product is the zero element. In this work, we will explore some zero divisor graph invariants constructed on the rings of the form \(\mathfrak{R}=\mathbb{Z}_n,\) when \(n\) is a product of square free primes. In particular, we will find the radio number for zero-divisor graphs constructed on \(\mathbb{Z}_{\mathfrak{p}_1 \mathfrak{p}_2 \mathfrak{p}_3}\) where \(\mathfrak{p}_1, \mathfrak{p}_2,\) and \(\mathfrak{p}_3\) are distinct primes with \(2 \leq \mathfrak{p}_3 < \mathfrak{p}_2 < \mathfrak{p}_1\) and combining with the known results for \(\mathbb{Z}_{\mathfrak{p}^3}\) and \(\mathbb{Z}_{\mathfrak{p}_1^2 \mathfrak{p}_2}\), It covers all possible cases when \(n\) is divisible by at most three primes.

A graph \(G(V, E)\) is word-representable if there exists a word \(w\) over the alphabet \(V\) such that for distinct letters \(x,y\in V\), \(x\) and \(y\) alternate in \(w\) if and only if they are adjacent in \(G\). In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets \(X\) and \(Y\) such that the subgraphs induced by \(X\) and \(Y\) are cliques. Necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering are known. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most \(3\), and permutation graphs are the only co-bipartite graphs with representation number \(2\). We prove that the speed is at most \(2^{O(n \log n)}\) and the entropy is \(0\). In particular, we obtain an upper bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

Spread of information within a wide variety of systems can be represented as evolving processes on digraphs. Starting from a random subset of initially active vertices, the measures we introduce assess the probability, speed, or number of steps it takes to spread information to the entire digraph, thus achieving digraph synchrony. Some of these measures may be viewed as generalizations of digraph connectivity or as generalizations of the diameter of a digraph to higher-order diameters. The paper places considerable emphasis on the regular case of Cayley digraphs associated to finite groups. It is demonstrated that, with appropriate assumptions on the growth of the generating sets, all the higher-order diameters of random Cayley digraphs are almost surely at most 2, as the digraph order goes to infinity. Certain results on the velocity of spread of information in digraphs are also presented.

We introduce and study a new family of circulant digraphs associated with the cyclic group \({\mathbb Z}_N\), obtained by restricting admissible combinations of two generators \(a\) and \(b\) to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.