Tiling Edge-Ordered Graphs with Monotone Paths and Other Structures

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1808-1839, June 2024.
Abstract. Given graphs [math] and [math], a perfect [math]-tiling in [math] is a collection of vertex-disjoint copies of [math] in [math] that together cover all the vertices in [math]. The study of the minimum degree threshold forcing a perfect [math]-tiling in a graph [math] 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 [math]. In this paper we initiate the study of the analogous question for edge-ordered graphs. In particular, we characterize for which edge-ordered graphs [math] this problem is well-defined. We also apply the absorbing method to asymptotically determine the minimum degree threshold for forcing a perfect [math]-tiling in an edge-ordered graph, where [math] is any fixed monotone path.