A graph theory translation. The vertices of a connected graph $G$ can be colored with $2017$ colors (meaning that any edge connects vertices of different colors). Prove that there exists a coloring of $G$ with colors $1,2,\dots,2017$, such that for any two vertices $v,w$ of $G$ there exists a path between them $v=v_1\,v_2\,\dots\,v_n=w$, satisfying $v_i\,,\,v_{i+1}\,,i=1,2,\dots,n-1$ are colored with adjacent colors.

First we color $G$ with the colors $1,2,\dots,2017$. We start from a vertex $v_0$ of $G$ and let $V_1$ be the maximal set of vertices of $G$ we can reach starting at $v_0$ making some hops, every hop complying the rule the two ends of the edge(hop) being colored with adjacent colors. Denote $V_2=V(G)\setminus V_1$ that's, the set of all other vertices. The idea is to make some permutation of the colors only over $V_2$ , such that the result be a valid coloring of $G$ , but after that an expanding of $V_1$ be possible. Let $i,i+1$ be any two adjacent colors. Let $W_i, W_{i+1}$ be the vertices in $V_2$ colored respectively with $i$-th / $i+1$-th color. We recolor $W_i$ with color $i+1$ and $W_{i+1}$ with color $i$. After that, any two connected vertices of $V_2$ are still colored differently, because we just swap that two colors in $V_2$. Assume there are two vertices $v_1\in V_1\,,\, v_2\in V_2$ colored with the same color. Of course that color must be either $i$ ot $i+1$. But if so, the set $V_1$ wouldn't be a maximal one (before recoloring), since we could hop from $v_1$ to $v_2$ - they were colored with adjacent colors(before recoloring). Thus, the new coloring of $G$ is again a valid coloring. It is clear that after some swapping permutations of kind $(i,i+1)$ over $V_2$ we will reach a position there will be two vertices $v_1\in V_2, v_2\in V_2$ colored with adjacent colors. Hence, we can expand $V_1$ by at least adding $v_2$ to it. After some steps, we end up with $V_1$ be the entire set of vertices of $G$.

dgrozev wrote: The vertices of a connected graph $G$ can be colored with $2017$ colors (any two connected vertices are of different colors). Prove that there exists a coloring of $G$ with colors $1,2,\dots,2017$, such that any two vertices of $G$ can be connected by a path, which edges connect vertices of adjacent colors. The coloring required must satisfy the condition that any two connected vertices are of different colors too, right? Also, could you confirm whether "connected vertices" means pairs of vertices which can be connected via some path or pairs of vertices that there's an edge between them.

Sorry if there is something unclear, I made some corrections as much as I can. A coloring of a graph $G$ with some number of colors as usual means that any edge connects vertices of different colors. The refined coloring that one should prove existence of, is another coloring of $G$, but satisfying the additional condition every two vertices can be connected with a path any two consecutive vertices of are colored with adjacent colors. EDIT. Btw, I can confirm it as much as you can, since both of us only see the OP as it is. But spending some time pondering over the original statement, I am 100% sure that it is the exact meaning, since there are no possible other reasonable interpretation and this interpretation leads to a nice version.