Proper rainbow saturation for trees
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916434634866688 |
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| author | Lane, Andrew Morrison, Natasha |
| author_facet | Lane, Andrew Morrison, Natasha |
| contents | Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we initiate a systematic study of the proper rainbow saturation number for trees. We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. Our proofs reveal connections to the classical saturation and semi-saturation numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Proper rainbow saturation for trees Lane, Andrew Morrison, Natasha Combinatorics Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we initiate a systematic study of the proper rainbow saturation number for trees. We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. Our proofs reveal connections to the classical saturation and semi-saturation numbers. |
| title | Proper rainbow saturation for trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.15275 |