On the edge set of graphs of lattice paths

Abstract

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.