Let G be a connected graph on n ≥ 2 vertices with girth at least g such that the length of a maximal chain of successively adjacent vertices of degree 2 in G does not exceed k ≥ 1. Denote by u(G) the maximum number of leaves in a spanning tree of G. We prove that u(G) ≥ α g,k (υ(G) − k − 2) + 2 where \( {\alpha}_{g,1}=\frac{\left[\frac{g+1}{2}\right]}{4\left[\frac{g+1}{2}\right]+1} \) and \( {\alpha}_{g,k}=\frac{1}{2k+2} \) for k ≥ 2. We present an infinite series of examples showing that all these bounds are tight.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 62–73.
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Karpov, D.V. Lower Bounds on the Number of Leaves in Spanning Trees. J Math Sci 232, 36–43 (2018). https://doi.org/10.1007/s10958-018-3857-2
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DOI: https://doi.org/10.1007/s10958-018-3857-2