A Result on Large Induced Subgraphs with Prescribed Residues in Bipartite Graphs

Abstract

It was proved by Scott that for every $k\ge 2$, there exists a constant $c(k)>0$ such that for every bipartite $n$-vertex graph $G$ without isolated vertices, there exists an induced subgraph $H$ of order at least $c(k)n$ such that $\operatorname{deg}_H(v) \equiv 1\pmod{k}$ for each $v \in H$. Scott conjectured that $c(k) = \Omega(1/k)$, which would be tight up to the multiplicative constant. We confirm this conjecture.