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Abstract An edge‐coloured graph is said to berainbowif no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge‐coloured graph on vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of for this question. Very recently, Janzer–Sudakov and Kim–Lee–Liu–Tran independently removed the term in Tomon's bound, showing a bound of . We prove an upper bound of for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the term, and so, it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non‐abelian groups.
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An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs
Abstract We present an integer programming model to compute the strong rainbow connection number,src(G), of any simple graphG. We introduce several enhancements to the proposed model, including a fast heuristic, and a variable elimination scheme. Moreover, we present a novel lower bound forsrc(G) which may be of independent research interest. We solve the integer program both directly and using an alternative method based on iterative lower bound improvement, the latter of which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to 379 vertices.
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- Award ID(s):
- 1720225
- PAR ID:
- 10361049
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Networks
- Volume:
- 79
- Issue:
- 1
- ISSN:
- 0028-3045
- Page Range / eLocation ID:
- p. 3-19
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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