Abstract
Over recent years there has been much interest in both Turán 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 Kühn 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.
Highlights
Over recent years there has been interest in extending classical graph theory results to the setting of vertex ordered graphs
Space barriers yield Proposition 1.5 and give the lower bound in Cases (i) and (iv), divisibility barriers provide the lower bound for Case (ii) and local barriers provide the lower bound in Case (iii) – while we recall that for unordered graphs there are only space and divisibility barriers
The variety of extremal examples indicates that proving a sharp almost perfect H-tiling theorem for an arbitrary ordered graph H seems to be very difficult, we propose a general framework for obtaining such almost perfect tiling theorems
Summary
Over recent years there has been interest in extending classical graph theory results to the setting of vertex ordered graphs. We study the minimum degree required to ensure an ordered graph has a perfect Htiling In both the ordered and unordered settings, an H-tiling in a graph G is a collection of vertex-disjoint copies of H contained in G. A central result in the area is the Hajnal–Szemeredi theorem [14] from 1970, which characterises the minimum degree that ensures a graph contains a perfect Kr-tiling. Our main result resolves the problem for all ordered graphs H of interval chromatic number 2 Even in this restricted case the nature of the minimum degree threshold is diverse, with a range of extremal examples coming into play, including a construction which is neither a divisibility nor space barrier. A key property of the regularity method – which is regularly used to help embed (spanning) subgraphs in the unordered setting – breaks down for ordered graphs; we introduce an approach to overcome this
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