A monotone path in an edge‐ordered graph

Abstract

An edge‐ordered graph is an ordered pair (G, f), where G is a graph and f is a bijective function, f : E(G) → {1, 2, …, |E(G)|}. A monotone path of length k in (G, f) is a simple path Pk+1 : v1v2 … 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}) for i = 1, 2, …, k − 1.It is proved that a graph G has the property that (G, f) contains a monotone path of length three for every f iff G contains as a subgraph, an odd cycle of length at least five or one of six listed graphs.