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Rainbow Saturation
TL;DR: In this article, the authors introduce the notion of rainbow saturation and the corresponding rainbow saturation number, which is the saturation version of the rainbow Tur\'an numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstra\"ete.
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Abstract: We introduce a notion of rainbow saturation and the corresponding rainbow saturation number. This is the saturation version of the rainbow Tur\'an numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstra\"ete. We give examples of graphs for which the rainbow saturation number is bounded away from the ordinary saturation number. This includes all complete graphs $K_n$ for $n\geq 4$, and several bipartite graphs. It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow extremal number versus the traditional extremal number. We also show that saturation numbers are linear for a large class of graphs, providing a partial rainbow analogue of a well known theorem of K\'asonyi and Tuza. We conclude this paper with related open questions and conjectures.
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Citations
The rainbow saturation number is linear
Natalie C. Behague,Tom Johnston,Shoham Letzter,Natasha Morrison,Shannon Ogden +4 more
- 16 Nov 2022
TL;DR: In this paper , it was shown that for any non-empty graph H , the rainbow saturation number is linear in n , thus proving a conjecture of Gir˜ao, Lewis, and Popielarz.
Rainbow Tur\'an Methods for Trees
Victor Bradley Bednar,Neal Bushaw +1 more
- 25 Mar 2022
TL;DR: In this paper , the authors explore the reduction method for finding upper bounds on the rainbow Tur´an number, and use this to inform results for double stars, caterpillars, and perfect binary trees.
Proper Rainbow Saturation Numbers for Cycles
Anastasia Halfpap,Bernard Lidick'y,Tomás Masarík +2 more
TL;DR: Proper rainbow saturation numbers for cycles asymptotically determined. The study of rainbow Tur\'an number and proper rainbow saturation number is initiated.
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An extremal problem in graph theory
TL;DR: In this paper, the smallest integer such that there is a G (n;f0(n, k)) in which for every set of k vertices there is vertex joined to each of these.