All trees contain a large induced subgraph having all degrees 1 (mod k) Original Research Article

We prove that, for integers n ⩾ 2 and k ⩾ 2, every tree with n vertices contains an induced subgraph of order at least 2⌊(n + 2k − 3)/(2k − 1)⌋ with all degrees congruent to 1 modulo k. This extends a result of Radcliffe and Scott, and answers a question of Caro et al.