A long-standing conjecture asserts the existence of a positive constant c such that every simple graph of order n without isolated vertices contains an induced subgraph of order at least cn such that all degrees in this induced subgraph are odd. Radcliffe and Scott have proved the conjecture for trees, essentially with the constant c = 2/3. Scott proved a… (More)
We show that for n ~ 2, every n-vertex tree has a star forest of order exceeding n /2. There are many theorems in graph theory in which a graph is required to have a particular subgraph. A stronger requirement, however, is that it be an induced subgraph: i.e. a subgraph in which two vertices must be adjacent if they are adjacent in the big graph. A common… (More)
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, Krasikov and Roditty.