Monotone paths in edge-ordered sparse graphs

Abstract

An edge-ordered graph is an ordered pair ( G, f), where G= G( V, E) is a graph and f is a bijective function, f: E( G)→{1,2,…,| E( G)|}. f is called an edge ordering of G. A monotone path of length k in ( G, f) is a simple path P k+1 : v 1,v 2,…,v k+1 in G such that either, f(( v i , v i+1 ))< f(( v i+1 , v i+2 )) or f(( v i , v i+1 ))> f(( v i+1 , v i+2 )) for i=1,2,…, k−1. Given an undirected graph G, denote by α( G) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α( G) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.