A Hypergraph Version of a Graph Packing Theorem by Bollobás and Eldridge. Issue 2 (4th December 2012)

• Record Type: Journal Article
• Authors: Kostochka, Alexandr; Stocker, Christopher; Hamburger, Peter
• Is Part Of: Journal of graph theory. Volume 74:Issue 2(2013)
• Page Start: 222
• Page End: 235
• Publication Date: 2012-12-04
• Subjects: Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.21706
• Languages: English
• ISSNs: 0364-9024

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Two <italic>n</italic>‐vertex hypergraphs <italic>G</italic> and <italic>H</italic><italic>pack</italic>, if there is a bijection <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28hmp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mspace width="0.16em" /><mml:mo>:</mml:mo><mml:mspace width="0.16em" /><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo><mml:mo>→</mml:mo><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> such that for every edge <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28h19" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives>, the set <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28h0r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo><mml:mspace width="0.16em" /><mml:mo>:</mml:mo><mml:mspace width="0.16em" /><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>e</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives> is not an edge in <italic>H</italic>. Extending a theorem by Bollobás and Eldridge on graph packing to hypergraphs, we show that if <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28gzp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:math></alternatives> and <italic>n</italic>‐vertex hypergraphs <italic>G</italic> and <italic>H</italic> with <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28gx4" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mi>E</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:mo>+</mml:mo><mml:mo>|</mml:mo><mml:mi>E</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:mo>≤</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives> with no edges of size 0, 1, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28h9q" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives> and <italic>n</italic> do not pack, then either <list id="jgt21706-list-0001" list-type="roman-lower"><list-item><p>one of <italic>G</italic> and <italic>H</italic> contains a spanning graph‐star, and each vertex of the other is contained in a graph edge, or</p></list-item><list-item><p>one of <italic>G</italic> and <italic>H</italic> has <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28h85" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives> edges of size <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28h4z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives> not containing a given vertex, and for every vertex <italic>x</italic> of the other hypergraph some edge of size <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg37c28h2v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:03649024:media:jgt21706:jgt21706-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives> does not contain <italic>x</italic>.</p></list-item></list></p> </abstract>