Spanning Trees with Many Leaves1
نویسندگان
چکیده
A connected graph having large minimum vertex degree must have a spanning tree with many leaves. In particular, let l(n, k) be the maximum integer m such that every connected n-vertex graph with minimum degree at least k has a spanning tree with at least m leaves. Then l(n, 3) ≥ n/4 + 2, l(n, 4) ≥ (2n + 8) /5, and l(n, k) ≤ n − 3n/(k +1) + 2 for all k. The lower bounds are proved by an algorithm that constructs a spanning tree with at least the desired number of leaves. Finally, l(n, k) ≥ (1 − blnk/k)n for large k, again proved algorithmically, where b is any constant exceeding 2. 5. 1Most of this research was done while the authors visited the Institute for Mathematics and Its Applications at the University of Minnesota, Minneapolis, MN 55455. 2Research supported in part by NSF Grant DMS 86-06225 and Airforce Grant OSR-86-0076. 3Research supported in part by ONR Grant N00014-85K0570.
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