SPANNING TREES WITH FEW NON-LEAVES

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) >= n/k+1 (1- epsilon(k)) ln(k+ 1), where epsilon(k) tends to zero as k tends to infinity. Here we prove that f(n, k) <= n/k+1 (ln(k vertical bar 1)vertical bar 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) <= n vertical bar k+1 (ln(k + 1) + 0.5 root 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.