Long paths in the distance graphs in vector spases over finite fields

We study the following task. Let $E \subset \mathbb{F}_{q}^{d}$, be the subset of the $d-$ dimensional vector space over the finite field with $q$ elements. We define so-called distance graph of the set $E$ with distance equal to one. The distance between the vertices $x,y \in E$ is defined as follows $\parallel x - y \parallel=(x_{1}-y_{1})^{2}+\ldots +(x_{d}-y_{d})^{2}$. The vertices of the distance graph are the elements of $E$ and a pair of vertices $x,y \in E$ are connected by an edge if the distance between them is equal to one. In this paper long paths in the graph are studied. Namely the lower estimate for the length of the longest non-overlapping path in this graph is obtained. Under certain conditions, it is proved that the length of such path consists of the most of vertices of the set $E$. This complements the result from the paper of A. Iosevich and co-authors. In the proof we use some combinatoric arguments and results obtained by M. Rudnev and A. Iosevich and also joint result of M. Bennett, J. Chapman, D. Covert, D. Hart, A. Iosevich and J. Pakianathan. The main idea of constructing a long path in such a graph is as follows. We construct many paths of shorter length by standard methods. Further, based on the joint result of M.Rudnev and A. Iosevich on the distribution of distances between elements of the set $E$, we conclude that there exist a pair of vertices of two different paths with distance one. Thus, it is possible to connect some two paths already constructed for their vertices and get a longer path. This procedure is repeated iteratively until the path of the given length is constructed. We note that this method and the main result remains true for such distance graphs with any non-zero distance.