Motzkin posed the problem of finding the maximal density $\mu (M)$ of sets of integers in which the differences given by a set $M$ do not occur. The problem is already settled when $|M|\le 2$ or $M$ is a finite arithmetic progression. In this paper, we determine $\mu (M)$ when $M$ has some other structure. For example, we determine $\mu (M)$ when $M$ is a finite geometric progression.

For vertices $u,v$ in a connected graph $G$, a $u-v$ chordless path in $G$ is a $u-v$ monophonic path. The monophonic interval $J_G[u,v]$ consists of all vertices lying on some $u-v$ monophonic path in $G$. For $S\subseteq V(G)$, the set $J_G[S]$ is the union of all sets $J_G[u,v]$ for $u,v\in S$. A set $S\subseteq V(G)$ is a monophonic set of $G$ if $J_G[S]=V(G)$. The cardinality of a minimum monophonic set of $G$ is the monophonic number of $G$, denoted by $mn(G)$. In this paper, bounds for the monophonic number of the strong product graphs are obtained, and for several classes, improved bounds and exact values are obtained.

A hypergraph is a useful tool to model complex systems and can be considered a natural generalization of graphs. In this paper, we define some operations of fuzzy hypergraphs and strong fuzzy $r$-uniform hypergraphs, such as Cartesian product, strong product, normal product, lexicographic product, union, and join. We prove that if a hypergraph $H$ is formed by one of these operations, then this hypergraph is a fuzzy hypergraph or a strong fuzzy $r$-uniform hypergraph. Finally, we discuss an application of fuzzy hypergraphs.

Let $p_e(n)$ be the number of ways to make change for $n$ cents using pennies, nickels, dimes, and quarters. By manipulating the generating function for $p_e(n)$, we prove that the sequence $\{p_e(n)(\mathrm{mod}{\ell }^j)\}$ is periodic for every prime power $\ell$.

In 1972, Chvatal and Erdős showed that the graph $G$ with independence number $\alpha (G)$ no more than its connectivity $\kappa (G)$ (i.e., $\kappa (G)\ge \alpha (G)$) is hamiltonian. In this paper, we consider a kind of Chvatal and Erdős type condition on edge-connectivity $\lambda (G)$ and matching number (edge independence number). We show that if $\lambda (G)\ge {\alpha }^{\prime }(G)–1$, then $G$ is either supereulerian or in a well-defined family of graphs. Moreover, we weaken the condition $\kappa (G)\ge \alpha (G)–1$ in [11] to $\lambda (G)\ge \alpha (G)–1$ and obtain a similar characterization on non-supereulerian graphs. We also characterize the graph which contains a dominating closed trail under the assumption $\lambda (G)\ge {\alpha }^{\prime }(G)–2$.

The coloring number $col(G)$ of a graph $G$ is the smallest number $k$ for which there exists a linear ordering of the vertices of $G$ such that each vertex is preceded by fewer than $k$ of its neighbors. It is well known that $\chi (G)\le col(G)$ for any graph $G$, where $\chi (G)$ denotes the chromatic number of $G$. The Randić index $R(G)$ of a graph $G$ is defined as the sum of the weights $\frac{1}{\sqrt{d(u)d(v)}}$ of all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. We show that $\chi (G)\le col(G)\le 2R^{\prime }(G)\le R(G)$ for any connected graph $G$ with at least one edge, and $col(G)=2R^{\prime }(G)$ if and only if $G$ is a complete graph with some pendent edges attaching to its same vertex, where $R^{\prime }(G)$ is a modification of Randić index, defined as the sum of the weights $\frac{1}{max\{d(u),d(v)\}}$ of all edges $uv$ of $G$. This strengthens a relation between Randić index and chromatic number by Hansen et al. [7], a relation between Randić index and coloring number by Wu et al. [17] and extends a theorem of Deng et al. [2].

For any vertex $x$ in a connected graph $G$ of order $n\ge 2$, a set $S\subseteq V(G)$ is a $z$-detour monophonic set of $G$ if each vertex $v\in V(G)$ lies on a $x-y$ detour monophonic path for some element $y\in S$. The minimum cardinality of a $x$-detour monophonic set of $G$ is the $x$-detour monophonic number of $G$, denoted by $d{m}_z(G)$. An $x$-detour monophonic set $S_x$ of $G$ is called a minimal $x$-detour monophonic set if no proper subset of $S_x$ is an $x$-detour monophonic set. The upper $x$-detour monophonic number of $G$, denoted by $d{m}_x^{+}(G)$, is defined as the maximum cardinality of a minimal $x$-detour monophonic set of $G$. We determine bounds for it and find the same for some special classes of graphs. For positive integers $r,d,$ and $k$ with $2\le r\le d$ and $k\ge 2$, there exists a connected graph $G$ with monophonic radius $r$, monophonic diameter $d$, and upper $z$-detour monophonic number $k$ for some vertex $x$ in $G$. Also, it is shown that for positive integers $j,k,l,$ and $n$ with $2\le j\le k\le l\le n–7$, there exists a connected graph $G$ of order $n$ with $d{m}_x(G)=j$, $d{m}_x^{+}(G)=l$, and a minimal $x$-detour monophonic set of cardinality $k$.

Many authors define certain generalizations of the usual Fibonacci, Pell, and Lucas numbers by matrix methods and then obtain the Binet formulas and combinatorial representations of the generalizations of these number sequences. In this article, we firstly define and study the generalized Gaussian Fibonacci numbers and then find the matrix representation of the generalized Gaussian Fibonacci numbers and prove some theorems by these matrix representations.

Given two sets $A,B\subset {\mathbb{F}}_q$, of elements of the finite field ${\mathbb{F}}_q$, of $q$ elements, Shparlinski (2008) showed that the product set $\mathcal{A}\mathcal{B}=\{ab\mid a\in \mathcal{A},b\in \mathcal{B}\}$ contains an arithmetic progression of length $k\ge 3$ provided that \(k

3\) is the characteristic of $\mathbb{F}$, and $|\mathcal{A}||\mathcal{B}|\ge 2q^{2-1/(k-1)}$. In this paper, we recover Shparlinski’s result for the case of 3-term arithmetic progressions via spectra of product graphs over finite fields. We also illustrate our method in the setting of residue rings. Let $m$ be a large integer and $\mathbb{Z}/m\mathbb{Z}$ be the ring of residues mod $m$. For any two sets $\mathcal{A},\mathcal{B}\subset \mathbb{Z}/m\mathbb{Z}$ of cardinality $$|\mathcal{A}||\mathcal{B}|>m(\frac{r(m)m}{r(m{)}^{\frac{1}{2}}+1})$$, the product set $\mathcal{A}\mathcal{B}$ contains a $3$-term arithmetic progression, where $r(m)$ is the smallest prime divisor of $m$ and $r(m)$ is the number of divisors of $m$. The spectral proofs presented in this paper avoid the use of character and exponential sums, the usual tool to deal with problems of this kind.

A proper edge-coloring of a graph $G$ with colors $1,\dots ,t$ is called an interval $t$-coloring if the colors of edges incident to any vertex of $G$ form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, the least value of $t$ for which $G$ has an interval $t$-coloring is denoted by $w(G)$. A graph $G$ is outerplanar if it can be embedded in the plane so that all its vertices lie on the same (unbounded) face. In this paper, we show that if $G$ is a 2-connected outerplanar graph with $\mathrm{\Delta }(G)=3$, then $G$ is interval colorable and $$w(G)=\{\begin{array}{ll}3,& \text{if }|V(G)|\text{ is even},\\ 4,& \text{if }|V(G)|\text{ is odd}.\end{array}$$
We also give a negative answer to the question of Axenovich on the outerplanar triangulations.