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Abstract Alinear forestis a disjoint union of path graphs. Thelinear arboricity of a graph, denoted by , is the least number of linear forests into which the graph can be partitioned. Clearly, for any graph of maximum degree . For the upper bound, the long‐standingLinear Arboricity Conjecture(LAC) due to Akiyama, Exoo, and Harary from 1981 asserts that . A graph is apseudoforestif each of its components contains at most one cycle. In this paper, we prove thatthe union of any two pseudoforests of maximum degree up to 3 can be decomposed into three linear forests. Combining it with a recent result of Wdowinski on the minimum number of pseudoforests into which a graph can be decomposed, we prove that the LAC holds for the following simple graph classes: ‐degenerate graphs with maximum degree , all graphs on nonnegative Euler characteristic surfaces provided the maximum degree , and graphs on negative Euler characteristic surfaces provided the maximum degree , as well as graphs with no ‐minor satisfying some conditions on maximum degrees.
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This content will become publicly available on April 1, 2026
Essentially tight bounds for rainbow cycles in proper edge‐colourings
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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- Award ID(s):
- 2349013
- PAR ID:
- 10632324
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 130
- Issue:
- 4
- ISSN:
- 0024-6115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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