Tilings in vertex ordered graphs

TL;DR

This paper develops a framework for embedding perfect tilings in vertex ordered graphs, resolving the minimum degree threshold for all graphs with interval chromatic number 2, and introduces new methods in regularity and absorption.

Contribution

It introduces a general framework for perfect H-tilings in ordered graphs and solves the problem for graphs with interval chromatic number 2, expanding understanding beyond unordered cases.

Findings

Resolved the minimum degree threshold for perfect H-tilings in ordered graphs with interval chromatic number 2.

Developed novel approaches to regularity and absorbing methods.

Identified richer extremal examples compared to unordered graph scenarios.

Abstract

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. In the process of proving our results, novel approaches to both the regularity and absorbing methods are developed.