Monotone paths in edge-ordered sparse graphs

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 Pk+1: v1,v2,…,vk+1 in G such that either, f((vi,vi+1))f((vi+1,vi+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.