This note explores a new family of graphs defined on the set of paths of the m × n lattice. We let each of the paths of the lattice be represented by a vertex, and connect two vertices by an edge if the corresponding paths share more than k steps, where k is a fixed parameter 0 = k = m + n . Each such graph is denoted by G ( m , n , k ) . Two large complete subgraphs of G ( m , n , k ) are described for all values of m , n , and k . The size of the edge set is determined for n = 2 , and a complicated recursive formula is given for the size of the edge set when k = 1 .