Abstract
Abstract
A recent paper of Balogh, Li and Treglown [3] 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 from [3] 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 Kühn and Osthus [Combinatorica 2009] and of Komlós [Combinatorica 2000]. Each of our results exhibits some curious, and perhaps unexpected, behaviour. Our solution to (i) makes use of a novel absorbing argument.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Cited by
4 articles.
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2. On Perfect Subdivision Tilings;Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications;2023
3. Tiling Dense Hypergraphs;Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications;2023
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