A Cop and Drunken Robber (CDR) game is one variation of a famous combinatorial game, called Cops and Robbers, which has been extensively studied and applied in the area of theoretical and computer science as demonstrated by several conferences and publications. In this paper, for a natural number n, we present two strategies for a single cop to chase a drunken robber on n-dimensional infinite-grid graphs. Both strategies show that if the initial distance between the cop and the drunken robber is s, then the expected capture time is $s+o\left(s\right)$.

For positive integers $n$ and $r$ such that $r \leq \lfloor n/2\rfloor$, let $X$ be a set of $n$ elements and let $\binom{X}{r}$ be the family of all $r$-subsets of $X$. Two sub-families $\mathcal{A}$ and $\mathcal{B}$ of $\binom{X}{r}$ are called cross intersecting if $A \cap B \neq \emptyset$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$. One of main tools in the study of extremal set theory, and cross intersecting families in particular, is compression operation. In this paper, we give an example of cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ that the compression operation is not applicable when $\mathcal{A}$ and $\mathcal{B}$ are disjoint. We develop new technique to prove that, for disjoint cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ of $\binom{X}{r}$, $|\mathcal{A}| + |\mathcal{B}| \leq \binom{n}{r} - \binom{l}{p}$ where $n = 2r + l$ and $p = min\{r, \lceil \frac{l}{2}\rceil\}$. This bound is asymptotically sharp.

Counting the number of maximal independent sets of graphs was started over $50$ years ago by Erd\H{o}s and Mooser. The problem has been continuously studied with a number of variations. Interestingly, when the maximal condition of an independent set is removed, such the concept presents one of topological indices in molecular graphs, the so called Merrifield-Simmons index. In this paper, we applied the concept of bivariate generating function to establish the recurrence relations of the numbers of maximal independent sets of regualr $n$-gonal cacti when $3 \leq n \leq 6$. By the ideas on meromorphic functions and the growth of power series coefficients, the asymptotic behaviors through simple functions of these recurrence relations have been established.

The angel game is played on $2$-dimensional infinite grid by $2$ players, the angel and the devil. In each turn, the angel of power $c \in \mathbb{N}$ moves from her current point $(x, y)$ to a point $(x', y')$ which $\max\{|x - x'|, |y - y'|\} \leq c$ while the devil chooses a point to destroy in his turn. Then, the angel can no longer land on these destroyed points. The angel wins if she has a strategy to escape from the devil forever and the devil wins if he can cage the angel in his destroyed points by a finite number of turns. It was proved in 2007 that the angel of power at least $2$ always wins. In this paper, we rise the problem when the angel is drunk. She randomly moves to any point in the range of her power in each turn. In our game version, the devil must cage the angel by a given finite number of turns, otherwise, the angel wins. We present a strategy for the devil that: if the devil plays with this strategy, then for given $c \in \mathbb{N}$ and $\epsilon > 0$, the devil can cage the angel of power $c$ with probability greater than $1 - \epsilon$ if and only if the game is played on an $n$-dimensional infinite grid when $n \leq 2$. We also establish the results related to the hitting time once the angel is first time outside an $n$-dimensional sphere of a given radius. The numerical simulation results are also presented in the last section.

A cops and robbers game (CR) on graphs was originated in 1983 by Quilliot and by Nowakowski and Winkler independently. This game has been applied in many problems in the area of theoretical computer science such as information seeking, robot motion planning or network security as evidenced by many conferences and publications. The CR game has also been extensively studied and varied to many versions. In this paper, we focus on CR game under the condition that the robber performs a random walk. Namely, he drunkenly, or randomly, chooses his next move to escape the cop, while the cop plays optimally in order to minimize the expected drunk capture times $${\textit{dct}}(G)$$dct ( G ) of a graph G. We apply the concepts of expected hitting time in Markov Chain and combinatorial technique to find $${\textit{dct}}(G)$$dct ( G ) for graphs G that have been used in many applications which are cycles, complete multipartite graphs, grid graphs and prism graphs.