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Zap. Nauchn. Sem. POMI, 2011, Volume 391, Pages 18–34 (Mi znsl4566)  

This article is cited in 5 scientific papers (total in 5 papers)

Bounds of a number of leaves of spanning trees

A. V. Bankevicha, D. V. Karpovb

a Saint-Petersburg State University, Saint-Petersburg, Russia
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least $\frac14(s-2)+2$ leaves.
Let $G$ be a connected graph of girth $g$ with $v$ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph $G$ does not exceed $k\ge1$. We prove that $G$ has a spanning tree with at least $\alpha_{g,k}(v(G)-k-2)+2$ leaves, where $\alpha_{g,k}=\frac{[\frac{g+1}2]}{[\frac{g+1}2](k+3)+1}$ for $k<g-2$; $\alpha_{g,k}(v(G)-k-2)+2$ for $k\ge g-2$.
We present infinite series of examples showing that all these bounds are exact.

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

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English version:
Journal of Mathematical Sciences (New York), 2012, 184:5, 564–572

Document Type: Article
UDC: 519.172.1
Received: 15.09.2011

Citation: A. V. Bankevich, D. V. Karpov, “Bounds of a number of leaves of spanning trees”, Combinatorics and graph theory. Part III, Zap. Nauchn. Sem. POMI, 391, POMI, St. Petersburg, 2011, 18–34; J. Math. Sci. (N. Y.), 184:5 (2012), 564–572

Citation in format AMSBIB
\Bibitem{BanKar11}
\by A.~V.~Bankevich, D.~V.~Karpov
\paper Bounds of a~number of leaves of spanning trees
\inbook Combinatorics and graph theory. Part~III
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 391
\pages 18--34
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl4566}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2012
\vol 184
\issue 5
\pages 564--572
\crossref{https://doi.org/10.1007/s10958-012-0881-5}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884301886}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Bankevich, “Otsenki kolichestva visyachikh vershin v ostovnykh derevyakh v grafakh bez treugolnikov”, Kombinatorika i teoriya grafov. III, Zap. nauchn. sem. POMI, 391, POMI, SPb., 2011, 5–17  mathnet; Bankevich A. V., “Bounds of a number of leaves of spanning trees in graphs without triangles”, J. Math. Sci. (N. Y.), 184:5 (2012), 557–563  crossref
    2. D. V. Karpov, “Ostovnye derevya s bolshim kolichestvom visyachikh vershin: novye nizhnie otsenki cherez kolichestvo vershin stepenei 3 i ne menee 4”, Kombinatorika i teoriya grafov. V, Zap. nauchn. sem. POMI, 406, POMI, SPb., 2012, 31–66  mathnet  mathscinet; D. V. Karpov, “Spanning trees with many leaves: new lower bounds in terms of number of vertices of degree 3 and at least 4”, J. Math. Sci. (N. Y.), 196:6 (2014), 747–767  crossref
    3. D. V. Karpov, “Ostovnye derevya s bolshim kolichestvom visyachikh vershin: nizhnie otsenki cherez kolichestvo vershin stepenei 1, 3 i ne menee 4”, Kombinatorika i teoriya grafov. V, Zap. nauchn. sem. POMI, 406, POMI, SPb., 2012, 67–94  mathnet  mathscinet; D. V. Karpov, “Spanning trees with many leaves: lower bounds in terms of number of vertices of degree 1, 3 and at least 4”, J. Math. Sci. (N. Y.), 196:6 (2014), 768–783  crossref
    4. V. E. Alekseev, D. V. Zakharova, “Nezavisimye mnozhestva v grafakh bez podderevev s bolshim chislom listev”, Diskretn. analiz i issled. oper., 23:1 (2016), 5–16  mathnet  crossref  mathscinet  elib
    5. D. V. Karpov, “Nizhnie otsenki kolichestva listev v ostovnykh derevyakh”, Kombinatorika i teoriya grafov. VIII, Zap. nauchn. sem. POMI, 450, POMI, SPb., 2016, 62–73  mathnet  mathscinet
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