A blow-up lemma for approximate decompositions

Abstract

We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to long-standing decomposition problems. For instance, our results imply the following. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a quasi-random <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vertex graph and suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript s Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\dots ,H_s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are bounded degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vertex graphs with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript s Endscripts e left-parenthesis upper H Subscript i Baseline right-parenthesis less-than-or-equal-to left-parenthesis 1 minus o left-parenthesis 1 right-parenthesis right-parenthesis e left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> </mml:mrow> </mml:munderover> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{i=1}^{s} e(H_i) \leq (1-o(1)) e(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript s Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\dots ,H_s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be packed edge-disjointly into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the complete graph <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript n"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">K_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies an approximate version of the tree packing conjecture of Gyárfás and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemerédi’s regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy, and Szemerédi to the setting of approximate decompositions.