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Triangle-tilings in graphs without large independent sets. (English) Zbl 1391.05199

Summary: We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an \(n\)-vertex graph \(G\) with sublinear independence number. In this setting, we show that if \(\delta(G)\geq n/3+o(n)\), then \(G\) has a triangle-tiling covering all but at most four vertices. Also, for every \(r\geq 5\), we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that \(G\) is \(K_r\)-free and \(n\) is divisible by 3.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05B45 Combinatorial aspects of tessellation and tiling problems
05C07 Vertex degrees
05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)
05C35 Extremal problems in graph theory
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References:

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