# Bounds of the number of leaves of spanning trees

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We prove that every connected graph with *s* vertices of degree not 2 has a spanning tree with at least \( \frac{1}{4}\left( {s - 2} \right) \) + 2 leaves.

Let *G* be a connected graph of girth *g* with υ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph *G* does not exceed k ≥ 1. We prove that *G* has a spanning tree with at least α_{ g,k }(υ(*G*) − *k* − 2) + 2 leaves, where \( {\alpha_{g,k}} = \frac{{\left[ {\frac{{g + 1}}{2}} \right]}}{{\left[ {\frac{{g + 1}}{2}} \right]\left( {k + 3} \right) + 1}} \) for *k* < *g* − 2; \( {\alpha_{g,k}} = \frac{{g - 2}}{{\left( {g - 1} \right)\left( {k + 2} \right)}} \) for *k* ≥ *g* − 2.

We present infinite series of examples showing that all these bounds are tight. Bibliography: 12 titles.

## Keywords

Span Tree Connected Graph Adjacent Vertex Infinite Series Maximal Chain## Preview

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