Distribution of subtree sums

Abstract

Motivated by the conjectures of Bondy and Malkevitch on (almost-)pancyclicity of 4-connected planar graphs, we study the distribution of subtree weights in trees with positive integer vertex weights. Let T be a tree on N1 vertices and let c:V(T)→N be vertex weights such that ∑v∈V(T)c(v)=N2. Let TS(T;c) denote the set of weights of subtrees of T, and [N1,K]N≔{k∈N:N1≤k≤K}. We show that, for every integer N1≤K≤N2, if N2<2N1−1, we have |TS(T;c)∩[N1,K]N|≥K−N1+12. As a corollary, we show that, for every planar hamiltonian graph G with minimum degree at least 4 and for every integer K with N≤K≤|V(G)|, where N≔⌈|V(G)|2⌉+3, there exist K−N+12 distinct cycle lengths in [N,K]N.