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arXiv · 2409.15444

Improved bounds for proper rainbow saturation

Andrew Lane·Natasha Morrison
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Abstract

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.

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Andrew Lane, Natasha Morrison. 2024-09-23. Improved bounds for proper rainbow saturation. https://arxiv.org/abs/2409.15444

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