Monotone paths in edge-ordered sparse graphs (Q1841931)

An edge-ordered graph is an ordered pair \((G,f)\), where \(G=G(V,E)\) is a graph and \(f\) is a bijective function \(f:E(G)\to \{1,2,\dots,|E(G)|\}\). \(f\) is called an edge ordering of \(G\). A monotone path of length \(k\) in \((G,f)\) is a simple path \(P_{k+1}:v_1,v_2,\dots,v_{k+1}\) 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 \(i=1,2,\dots,k-1\). Given an undirected graph \(G\), denote by \(\alpha(G)\) the minimum over all edge orderings of the maximum length of a monotone path. The authors give bounds on \(\alpha(G)\) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity. For example, every planar graph \(G\) has \(\alpha(G)\leq 9\), and every bipartite planar graph \(G\) has \(\alpha(G)\leq 6\).