Spanning trees with few non-leaves

Abstract

Let f (n, k) denote the smallest number so that every connected graph with n vertices and minimum degree at least k contains a spanning tree in which the number of non-leaves is at most f (n, k). An early result of Linial and Sturtevant asserting that f (n, 3) = 3n/4 + O(1) and a related conjecture suggested by Linial led to a significant amount of work studying this function. It is known that for n much larger than k, $f(n,k) \ge {n \over {k + 1}}(1 - \varepsilon (k))\ln (k + 1)$, where ε(k) tends to zero as k tends to infinity. Here we prove that $f(n,k) \le {n \over {k + 1}}(\ln (k + 1) + 4) - 2$. This improves the error term in the best known upper bound for the function, due to Caro, West and Yuster, which is $f(n,k) \le {n \over {k + 1}}(\ln (k + 1) + 0.5\sqrt {\ln (k + 1)} + 145)$. The proof provides an efficient deterministic algorithm for finding such a spanning tree in any given input graph satisfying the assumptions.