A monotone path in an edge-ordered graph (Q1095926)

An edge-ordered graph is an ordered pair (G,f), where G is a graph and f is a bijective function, f: E(G)\(\to \{1,2,...,| E(G)| \}\). A monotone path of length k in (G,f) is a simple path \(P_{k+1}:\) \(v_ 1v_ 2...v_{k+1}\) in G such that either \(f(\{v_ i,v_{i+1}\})<f(\{v_{i+1},v_{i+2}\})\) or \(f(\{v_ i,v_{+1}\})>f(\{v_{i+1},v_ i\})\) 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.