A Zero-Sum Conjecture for Trees

We propose the following conjecture: Let \(m \geq k \geq 2\) be integers such that \(k \mid m\), and let \(T_m\) be a tree on \(m\) edges. Let \(G\) be a graph with \(\delta(G) \geq m+k-1\). Then for every \(Z_k\)-colouring of the edges of \(G\) there is a zero-sum (mod \(k\)) copy of \(T_m\) in \(G\). We prove the conjecture for \(m \geq k = 2\), and explore several relations to the zero-sum Turán numbers.