The pursuit-evasion game on the 1-skeleton graph of the regular polyhedron. I

We consider a game between a group of $n$ pursuers and one evader moving with the same maximal speed along 1-skeleton of a given regular polyhedron. The objective of the paper consists of finding an integer $N(M)$ possessing the following property: if $n \geq N(M)$ then the group of pursuers wins while if $n < N(M)$ then an evader wins. Part I of the paper is devoted to the case of polyhedrons in the space $\mathbb{R}^N$, Part II will be devoted to the case $\mathbb{R}^N$, $n\geq5$, and Part III will be devoted to the case $\mathbb{R}^4$.