Tiling edge-ordered graphs with monotone paths and other structures

Given graphs 𝐹 and 𝐺, a perfect 𝐹-tiling in 𝐺 is a collection of vertex-disjoint copies of 𝐹 in 𝐺 that together cover all the vertices in 𝐺. The study of the minimum degree threshold forcing a perfect 𝐹-tiling in a graph 𝐺 has a long history, culminating in the Kühn–Osthus theorem [D. Kühn and D. Osthus, Combinatorica, 29 (2009), pp. 65–107] which resolves this problem, up to an additive constant, for all graphs 𝐹. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs 𝐹 this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect 𝑃-tiling in an edge-ordered graph, where 𝑃 is any fixed monotone path.