Tilings in vertex ordered graphs

Jozsef Balogh, Lina Li, Andrew Treglown

Published 2020-07-21Version 1

Over recent years there has been much interest in both Tur\'an and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In the (unordered) graph setting, this problem was resolved by K\"uhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009]. We use our general framework to resolve the perfect $H$-tiling problem for all ordered graphs $H$ of interval chromatic number $2$. Already in this restricted setting the class of extremal examples is richer than in the unordered graph problem.