The lazy cop number is the minimum number of cops needed for the cops to have a winning strategy in the game of Cops and Robbers where at most one cop may move in any round. This variant of the game of Cops and Robbers, called Lazy Cops and Robbers, was introduced by Offner and Ojakian, who provided bounds for the lazy cop number of the hypercube. In this paper, we are interested in the game of Lazy Cops and Robbers on generalized hypercubes. Generalizing existing methods, we will give asymptotic upper and lower bounds on the lazy cop number of the generalized hypercube.

In 1977, Davis et al. proposed a method to generate an arrangement of $\left[n\right]=\{1,2,\dots ,n\}$ that avoids three-term monotone arithmetic progressions. Consequently, this arrangement avoids k-term monotone arithmetic progressions in $\left[n\right]$ for $k\ge 3$. Hence, we are interested in finding an arrangement of $\left[n\right]$ that avoids k-term monotone arithmetic progression, but allows $k-1$-term monotone arithmetic progression. In this paper, we propose a method to rearrange the rows of a magic square of order $2k-3$ and show that this arrangement does not contain a k-term monotone arithmetic progression. Consequently, we show that there exists an arrangement of n consecutive integers such that it does not contain a k-term monotone arithmetic progression, but it contains a $k-1$-term monotone arithmetic progression.

By recalling van der Waerden theorem, there exists a least a positive integer $w=w(k;r)$ such that for any $n\ge w$, every r-colouring of $[1,n]$ admits a monochromatic k-term arithmetic progression. Let $k\ge 2$ and $r_k\left(n\right)$ denote the minimum number of colour required so that there exists a $r_k\left(n\right)$-colouring of $[1,n]$ that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for $r_k(n+1)=r_k\left(n\right)$. We also show that $r_k\left(n\right)=2$ for all $k\le n\le 2{(k-1)}^2$ and give an upper bound for $r_p\left(p^m\right)$ for any prime $p\ge 3$ and integer $m\ge 2$.

Let p be a prime and ${\mathbb{F}}_p$ be the set of integers modulo p. Let ${\chi }_p$ be a function defined on ${\mathbb{F}}_p$ such that ${\chi }_p\left(0\right)=0$ and for $a\in {\mathbb{F}}_p\backslash \left\{0\right\}$, set ${\chi }_p\left(a\right)=1$ if a is a quadratic residue modulo p and ${\chi }_p\left(a\right)=-1$ if a is a quadratic non-residue modulo p. Note that ${\chi }_p\left(a\right)=\left({\displaystyle \frac{a}{p}}\right)$ is indeed the Legendre symbol. The image of ${\chi }_p$ in the set of real numbers. In this paper, we consider the following sum ${\sum }_{x\in {\mathbb{F}}_p}{\chi }_p((x-a_1)(x-a_2)\dots (x-a_t))$ where $a_1,a_2,\dots ,a_t$ are distinct elements in ${\mathbb{F}}_p$.

Graph burning is a discrete-time deterministic process that can be interpreted as a model for spreading influence in social networks. The burning number conjecture says that the burning number of any connected graph of order $$m^2$$ m 2 is at most m and was proven to hold asymptotically recently. Since the early days, spiders have garnered a significant amount of attention. Not only were spiders shown to satisfy the conjecture, but also determining the burning numbers for spiders was an NP-complete problem. Furthermore, Tan and Teh obtained a tight upper bound beyond $$m^2$$ m 2 on the order of a spider to guarantee its burnability in m rounds, and, surprisingly, it depends only on the number of arms, with balanced spiders as exceptions for small m. Inspired by these works, we first study the largest attainable order for balanced spiders, given the number of arms and the burning number. The second half of this work lays the foundation for generalization to all trees. As highlights of this work, we obtain some principal general properties on the associated neighborhoods corresponding to any optimal burning sequence of the extremal trees with a given burning number. Finally, we apply the principles to determine the orders of the extremal trees with two branch vertices and conclude by discussing how to extend our study.