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Cycles Are Strongly Ramsey-Unsaturated

Published online by Cambridge University Press:  29 April 2014

J. SKOKAN  and
M. STEIN
J. SKOKAN
Affiliation:
Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK and Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA (e-mail: [email protected])
M. STEIN*
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile (e-mail: [email protected])
*
‡Corresponding author.
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Abstract

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [J. Graph Theory51 (2006), pp. 22–32], where it is shown that cycles (except for C4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn, unless n is even and adding the chord creates an odd cycle.

We prove this conjecture for large cycles by showing a stronger statement. If a graph H is obtained by adding a linear number of chords to a cycle Cn, then r(H)=r(Cn), as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large.

This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method.

Keywords

Primary 05C55Secondary 05D1005C38
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 4 , July 2014 , pp. 607 - 630
Copyright
Copyright © Cambridge University Press 2014 

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