Abstract
Let L(n, d) denote the minimum possible number of leaves in a tree of order n and diameter d. Lesniak (1975) gave the lower bound B(n,d) = ⌈2(n − 1)/d⌉ for L(n,d). When d is even, B(n,d) = L(n,d). But when d is odd, B(n,d) is smaller than L(n,d) in general. For example, B(21, 3) = 14 while L(21, 3) = 19. In this note, we determine L(n, d) using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.
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L. Lesniak: On longest paths in connected graphs. Fundam. Math. 86 (1975), 283–286.
O. Ore: Theory of Graphs. Colloquium Publications 38. American Mathematical Society, Providence, 1962.
D. B. West: Introduction to Graph Theory. Prentice Hall, Upper Saddle River, 1996.
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The authors were supported by the NSFC grants 11671148 and 11771148 and Science and Technology Commission of Shanghai Municipality (STCSM) grant 18dz2271000.
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Qiao, P., Zhan, X. The relation between the number of leaves of a tree and its diameter. Czech Math J 72, 365–369 (2022). https://doi.org/10.21136/CMJ.2021.0492-20
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DOI: https://doi.org/10.21136/CMJ.2021.0492-20