Asymptotically optimal neighbour sum distinguishing colourings of graphs1. Issue 4 (9th June 2014)

• Record Type: Journal Article
• Authors: Przybyło, Jakub
• Is Part Of: Random structures & algorithms. Volume 47:Issue 4(2015)
• Page Start: 776
• Page End: 791
• Publication Date: 2014-06-09
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20553
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Consider a simple graph <italic>G</italic> = (<italic>V, E</italic>) and its <italic>proper</italic> edge colouring <italic>c</italic> with the elements of the set {1, 2, …, <italic>k</italic>}. The colouring <italic>c</italic> is said to be <italic>neighbour sum distinguishing</italic> if for every pair of vertices <italic>u, v</italic> adjacent in <italic>G</italic>, the sum of colours of the edges incident with <italic>u</italic> is distinct from the corresponding sum for <italic>v</italic>. The smallest integer <italic>k</italic> for which such colouring exists is known as the <italic>neighbour sum distinguishing index</italic> of a graph and denoted by <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk524" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20553:rsa20553-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>χ</mml:mo><mml:msub><mml:mo>′</mml:mo><mml:mo>∑</mml:mo></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. The definition of this parameter, which makes sense for graphs containing no isolated edges, immediately implies that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk4z3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20553:rsa20553-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>χ</mml:mo><mml:msub><mml:mo>′</mml:mo><mml:mo>∑</mml:mo></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mo>Δ</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, where Δ is the maximum degree of <italic>G</italic>. On the other hand, it was conjectured by Flandrin et al. that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk4w2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20553:rsa20553-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>χ</mml:mo><mml:msub><mml:mo>′</mml:mo><mml:mo>∑</mml:mo></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mo>Δ</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> for all those graphs, except for <italic>C</italic><sub>5</sub>. We prove this bound to be asymptotically correct by showing that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvk4xk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20553:rsa20553-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>χ</mml:mo><mml:msub><mml:mo>′</mml:mo><mml:mo>∑</mml:mo></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mo>Δ</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:mrow></mml:math></alternatives></inline-formula>. The main idea of our argument relays on a random assignment of the colours, where the choice for every edge is biased by so called <italic>attractors</italic>, randomly assigned to the vertices. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 776–791, 2015</p> </abstract>