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arXiv · 1111.3266

Bounds of a number of leaves of spanning trees

Anton Bankevich·Dmitri Karpov
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Abstract

We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\ge 1$. We prove that $G$ has a spanning tree with at least $α_{g,k}(v(G)-k-2)+2$ leaves, where $α_{g,k}= {[{g+1\over2}]\over [{g+1\over2}](k+3)+1}$ for $k<g-2$; $α_{g,k}= {g-2\over (g-1)(k+2)}$ for $k\ge g-2$. We present infinite series of examples showing that all these bounds are exact.

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Anton Bankevich, Dmitri Karpov. 2012-06-18. Bounds of a number of leaves of spanning trees. https://doi.org/10.1007/s10958-012-0881-5

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