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 Zap. Nauchn. Sem. POMI, 2016, Volume 450, Pages 62–73 (Mi znsl6337)

Lower bounds on the number of leaves in spanning trees

D. V. Karpovab

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
b St. Petersburg State University, St. Petersburg, Russia

Abstract: Let $G$ be a connected graph on $n\ge2$ vertices with girth at least $g$. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\ge1$. Denote by $u(G)$ the maximal number of leaves in a spanning tree of $G$. We prove, that $u(G)\ge\alpha_{g,k}(v(G)-k-2)+2$, where $\alpha_{g,1}=\frac{[\frac{g+1}2]}{4[\frac{g+1}2]+1}$ and $\alpha_{g,k}=\frac1{2k+2}$ for $k\ge2$. We present infinite series of examples showing that all these bounds are tight.

Key words and phrases: spanning tree, number of leaves.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 14.Z50.31.0030ÍØ-9721.2016.1 Russian Foundation for Basic Research 14-01-00156

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English version:
Journal of Mathematical Sciences (New York), 2018, 232:1, 36–43

Bibliographic databases:

UDC: 519.172.1

Citation: D. V. Karpov, “Lower bounds on the number of leaves in spanning trees”, Combinatorics and graph theory. Part VIII, Zap. Nauchn. Sem. POMI, 450, POMI, St. Petersburg, 2016, 62–73; J. Math. Sci. (N. Y.), 232:1 (2018), 36–43

Citation in format AMSBIB
\Bibitem{Kar16} \by D.~V.~Karpov \paper Lower bounds on the number of leaves in spanning trees \inbook Combinatorics and graph theory. Part~VIII \serial Zap. Nauchn. Sem. POMI \yr 2016 \vol 450 \pages 62--73 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6337} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3582953} \transl \jour J. Math. Sci. (N. Y.) \yr 2018 \vol 232 \issue 1 \pages 36--43 \crossref{https://doi.org/10.1007/s10958-018-3857-2} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85047306988}