Dirac-type results for tilings and coverings in ordered graphs

TL;DR

This paper establishes asymptotic minimum degree thresholds for tilings and coverings in ordered graphs, extending classical tiling theorems to the ordered setting and resolving several open questions.

Contribution

It provides the first asymptotic results for Dirac-type tiling problems in ordered graphs, including perfect and almost perfect tilings, using novel absorbing techniques.

Findings

Determined thresholds for perfect $H$-tilings in ordered graphs.

Established thresholds for partial $H$-tilings covering a fixed proportion.

Resolved open questions by Balogh, Li, Treglown, and Falgas-Ravry.

Abstract

A recent paper of Balogh, Li and Treglown initiated the study of Dirac-type problems for ordered graphs. In this paper we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for forcing (i) a perfect $H$-tiling in an ordered graph, for any fixed ordered graph $H$ of interval chromatic number at least $3$; (ii) an $H$-tiling in an ordered graph $G$ covering a fixed proportion of the vertices of $G$ (for any fixed ordered graph $H$); (iii) an $H$-cover in an ordered graph (for any fixed ordered graph $H$). The first two of these results resolve questions of Balogh, Li and Treglown whilst (iii) resolves a question of Falgas-Ravry. Note that (i) combined with a result of Balogh, Li and Treglown completely determines the asymptotic minimum degree threshold for forcing a perfect $H$-tiling. Additionally, we prove a result that combined with a theorem of Balogh, Li and Treglown, asymptotically determines the minimum degree threshold for forcing an almost perfect $H$-tiling in an ordered graph (for any fixed ordered graph $H$). Our work therefore provides ordered graph analogues of the seminal tiling theorems of K\"uhn and Osthus [Combinatorica 2009] and of Koml\'os [Combinatorica 2000]. Each of our results exhibits some curious, and perhaps unexpected, behaviour. Our solution to (i) makes use of a novel absorbing argument.