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- W4387402806 abstract "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 $23$." @default.
- W4387402806 created "2023-10-07" @default.
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- W4387402806 date "2023-10-06" @default.
- W4387402806 modified "2023-10-07" @default.
- W4387402806 title "Improved Pyrotechnics: Closer to the Burning Number Conjecture" @default.
- W4387402806 doi "https://doi.org/10.37236/11113" @default.
- W4387402806 hasPublicationYear "2023" @default.
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