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Spanning trees in graphs of minimum degree 4 or 5. (English) Zbl 0776.05031
Summary: For a connected simple graph \(G\) let \(L(G)\) denote the maximum number of leaves in any spanning tree of \(G\). Lineal conjectured that if \(G\) has \(N\) vertices and minimum degree \(k\), then \(L(G)\geq((k-2)/(k+1))N+c_ k\), where \(c_ k\) depends on \(k\). We prove that if \(k=4\), \(L(G)\geq{2\over 5}N+{8\over 5}\); if \(k=5\), \(L(G)\geq{1\over 2}N+2\). We give examples showing that these bounds are sharp.

MSC:
05C05 Trees
05C35 Extremal problems in graph theory
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