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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.15444 |
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| _version_ | 1866912071087554560 |
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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 use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on $\text{sat}^*(n,H)$ for general cliques, cycles, and complete bipartite graphs. We also provide some general lower bounds on $\text{sat}^*(n,H)$ and explore several other interesting directions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved bounds for proper rainbow saturation 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 use connections to the classical saturation and semi-saturation numbers to provide new upper bounds on $\text{sat}^*(n,H)$ for general cliques, cycles, and complete bipartite graphs. We also provide some general lower bounds on $\text{sat}^*(n,H)$ and explore several other interesting directions. |
| title | Improved bounds for proper rainbow saturation |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.15444 |