تقرير
The rainbow saturation number is linear
| العنوان: | The rainbow saturation number is linear |
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| المؤلفون: | Behague, Natalie, Johnston, Tom, Letzter, Shoham, Morrison, Natasha, Ogden, Shannon |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics |
| الوصف: | Given a graph $H$, we say that an edge-coloured graph $G$ is $H$-rainbow saturated if it does not contain a rainbow copy of $H$, but the addition of any non-edge in any colour creates a rainbow copy of $H$. The rainbow saturation number $\text{rsat}(n,H)$ is the minimum number of edges among all $H$-rainbow saturated edge-coloured graphs on $n$ vertices. We prove that for any non-empty graph $H$, the rainbow saturation number is linear in $n$, thus proving a conjecture of Gir\~{a}o, Lewis, and Popielarz. In addition, we also give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Gir\~{a}o, Lewis, and Popielarz. Comment: 16 pages, 2 figures |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1137/23M1566881 |
| URL الوصول: | http://arxiv.org/abs/2211.08589 |
| رقم الأكسشن: | edsarx.2211.08589 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1137/23M1566881 |
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