Abstract:A graph $G$ admits an $H$-tiling if it contains a collection of vertex-disjoint copies of $H$. In this paper, we confirm a conjecture proposed by Kühn, Osthus, and Treglown by showing that for any given graph $H$, there exists a constant $C(H)$ such that the following holds. If $G$ is a sufficiently large $n$-vertex graph satisfying $d(x) + d(y) \geq 2\left(1 - 1/\chi_{\text{cr}}(H)\right)n$ for all nonadjacent vertices $x, y \in V(G)$, then $G$ contains an $H$-tiling covering all but at most $C(H)$ vertices. Here $\chi_{\text{cr}}(H)$ denotes the critical chromatic number of $H$.
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.22553 [math.CO] |
| (or arXiv:2605.22553v2 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.22553 arXiv-issued DOI via DataCite |
From: Linpeng Zhang [view email]
[v1]
Thu, 21 May 2026 14:35:30 UTC (31 KB)
[v2]
Fri, 22 May 2026 02:38:15 UTC (31 KB)
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