We call a graph with n vertices $k-flowing-chromatic$ if:
1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors.
2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors.
3. after some action of step 2 we can make all the chess reach each of the n vertices.
Let T(G) denote the least number k such that G is k-flowing-chromatic.
If such k does not exist, denote T(G)=0.
denote $\chi (G)$ the chromatic number of G.
Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.
1. **Case \( m = 1 \)**:
   - For \( m = 1 \), the graph \( G \) must be a single vertex, as any graph with more than one vertex and \( \chi(G) = 1 \) would not be connected. 
   - Since \( G \) is a single vertex, it trivially satisfies the conditions for being \( k \)-flowing-chromatic with \( k = 1 \). 
   - Therefore, \( T(G) = 1 \) for \( m = 1 \).

2. **Case \( m = 2 \)**:
   - For \( m = 2 \), \( G \) is a bipartite graph. 
   - A bipartite graph can be colored with 2 colors, so \( \chi(G) = 2 \).
   - However, for \( T(G) \ge 2^m = 4 \), we need to ensure that the graph has a Hamiltonian cycle and satisfies the conditions for \( k \)-flowing-chromatic with \( k \ge 4 \).
   - Bipartite graphs with a Hamiltonian cycle can be 2-colored, but they cannot satisfy \( T(G) \ge 4 \) because they cannot be 4-flowing-chromatic.
   - Thus, \( m \leq 2 \) fails.

3. **Constructing a graph \( G \) for \( m = 3 \)**:
   - We need to construct a graph \( G \) such that \( \chi(G) = 3 \), \( G \) has a unique Hamiltonian cycle, and \( T(G) \ge C \) for any positive integer \( C \).
   - Start by picking a sufficiently large integer \( N \) and construct a cycle \( C_N \) with \( N \) vertices. Label the vertices counterclockwise as \( u_1, u_2, \ldots, u_N \).
   - Define an integer sequence: 
     \[
     x_1 = 1, \quad x_{t+1} = x_t + 2018 + t, \quad 1 \leq t \leq D-1
     \]
     where \( D \) is a sufficiently large integer (Note that we need \( N \) to be larger than \( x_D \)).
   - Connect edges \( u_{x_i}u_{x_{i+1}}, 1 \leq i \leq D-1 \).
   - This construction ensures that the graph \( G \) has a unique Hamiltonian cycle and can be 3-colored.
   - By choosing \( D \ge 10000C \), we ensure that \( T(G) \ge C \).

4. **Verification**:
   - The graph \( G \) constructed has \( \chi(G) = 3 \) because it is a cycle with additional edges that do not create any smaller cycles.
   - The unique Hamiltonian cycle is the original cycle \( C_N \).
   - The additional edges ensure that the graph can be \( k \)-flowing-chromatic for large \( k \), satisfying \( T(G) \ge C \).

The final answer is \( \boxed{ m = 3 } \).