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)) or 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.