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Article Dans Une Revue The Electronic Journal of Combinatorics Année : 2023

Improved Pyrotechnics: Closer to the Burning Number Conjecture

Résumé

The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n}\, \rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order of the graph is much larger than its growth, which is the maximal distance of a vertex to a well-chosen path in the graph. We prove that the conjecture for graphs of bounded growth reduces to a finite number of cases. We provide the best-known bound on the burning number of a connected graph $G$ of order $n,$ given by $b(G) \le \sqrt{4n/3} + 1,$ improving on the previously known $\sqrt{3n/2}+O(1)$ bound. Using the improved upper bound, we show that the conjecture almost holds for all graphs with minimum degree at least $3$ and holds for all large enough graphs with minimum degree at least $4$. The previous best-known result was for graphs with minimum degree $23$.

Dates et versions

hal-04587525 , version 1 (24-05-2024)

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Citer

Paul Bastide, Marthe Bonamy, Anthony Bonato, Pierre Charbit, Shahin Kamali, et al.. Improved Pyrotechnics: Closer to the Burning Number Conjecture. The Electronic Journal of Combinatorics, 2023, 30 (4), ⟨10.37236/11113⟩. ⟨hal-04587525⟩
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