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Generalised Ramsey numbers and Bruhat order on involutionsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Linköping: Linköping University Electronic Press, 2015. , 14 p.
##### Series

Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1734
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:liu:diva-123044DOI: 10.3384/lic.diva-123044ISBN: 978-91-7685-892-9 (print)OAI: oai:DiVA.org:liu-123044DiVA: diva2:876270
##### Presentation

2015-12-17, Nobel (BL32), ingång 23, B-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (Swedish)
##### Opponent

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##### Supervisors

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#####

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Available from: 2015-12-03 Created: 2015-12-03 Last updated: 2015-12-03Bibliographically approved
##### List of papers

This thesis consists of two papers within two different areas of combinatorics.

Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs. More precisely, we study generalised Ramsey numbers for two sets Γ_{1} and Γ_{2} of cycles. We determine, in particular, all generalised Ramsey numbers R(Γ_{1}, Γ_{2}) such that Γ_{1} or Γ_{2} contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also characterise many (Γ_{1}, Γ_{2})-critical graphs. As special cases, we obtain complete characterisations of all (C_{n},C_{3})-critical graphs for n ≥ 5, and all (C_{n},C_{5})-critical graphs for n ≥ 6.

In Paper B, we study the combinatorics of certain partially ordered sets. These posets are unions of conjugacy classes of involutions in the symmetric group S_{n}, with the order induced by the Bruhat order on S* _{n}*. We obtain a complete characterisation of the posets that are graded. In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative. When the posets are graded, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.

1. Generalised Ramsey numbers for two sets of cycles$(function(){PrimeFaces.cw("OverlayPanel","overlay876248",{id:"formSmash:j_idt423:0:j_idt427",widgetVar:"overlay876248",target:"formSmash:j_idt423:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. The Bruhat order on conjugation-invariant sets of involutions in the symmetric group$(function(){PrimeFaces.cw("OverlayPanel","overlay876258",{id:"formSmash:j_idt423:1:j_idt427",widgetVar:"overlay876258",target:"formSmash:j_idt423:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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