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Generalised Ramsey numbers for two sets of cyclesPrimeFaces.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}); (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### Keyword [en]

Generalised Ramsey number, critical graph, cycle, set of cycles
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:liu:diva-123042OAI: oai:DiVA.org:liu-123042DiVA: diva2:876248
#####

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Available from: 2015-12-03 Created: 2015-12-03 Last updated: 2015-12-03Bibliographically approved
##### In thesis

We determine several generalised Ramsey numbers for two sets Γ_{1} and Γ_{2} of cycles, 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 from the 1970s. Notably, including both C_{3} and C_{4} in one of the sets, makes very little difference from including only C_{4}. Furthermore, we give a conjecture for the general case. We also describe many (Γ_{1}, Γ_{2})-avoiding graphs, including a complete characterisation of most (Γ_{1}, Γ_{2})-critical graphs, i.e., (Γ_{1}, Γ_{2})-avoiding graphs on R(Γ_{1}, Γ_{2}) − 1 vertices, such that Γ_{1} or Γ_{2} contains a cycle of length at most 5. For length 4, this is an easy extension of a recent result of Wu, Sun, and Radziszowski, in which |Γ_{1}| = |Γ_{2}| = 1. For lengths 3 and 5, our results are new even in this special case.

1. Generalised Ramsey numbers and Bruhat order on involutions$(function(){PrimeFaces.cw("OverlayPanel","overlay876270",{id:"formSmash:j_idt647:0:j_idt651",widgetVar:"overlay876270",target:"formSmash:j_idt647:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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