On the edge set of graphs of lattice paths

Steven Klee,Rick Gillman,Lara Pudwell

doi:10.1155/s0161171204306058

Abstract

This note explores a new family of graphs defined on the set of paths of them×nlattice. 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 thanksteps, wherekis a fixed parameter0=k=m+n. Each such graph is denoted byG(m,n,k). Two large complete subgraphs ofG(m,n,k)are described for all values ofm,n, andk. The size of the edge set is determined forn=2, and a complicated recursive formula is given for the size of the edge set whenk=1.

Highlights

A classic combinatorial problem, presented in nearly every introductory text, is enumerating the number of distinct paths on an m × n rectangular lattice
Following Gillman [2], we say two paths are essentially the same, or (k + 1)-. Equivalent, if they share more than k steps
We let C denote m+n n and will let {P1, P2, . . . , PC } denote the set of all paths on the m × n lattice, with paths listed in reverse lexicographic order

Summary

Introduction

A classic combinatorial problem, presented in nearly every introductory text, is enumerating the number of distinct paths on an m × n rectangular lattice. For the purposes of this note, we let m denote the number of rows and n denote the number of columns of rectangular cells in the lattice. PC } denote the set of all paths on the m × n lattice, with paths listed in reverse lexicographic order. The set of all paths on the m × n lattice, denoted as L(m, n), can be viewed as the vertices of a graph. This graph is denoted as G(m, n, k), and its edge set is E(m, n, k). In L(m, n), the paths of the forms Nk+1 En Nm−(k+1) and Ek+1 Nm En−(k+1) are called barrier paths and denoted as Pyk and Pxk , respectively, when the paths of the m × n lattice are listed in reverse lexicographic order

Result

Conclusion