A note 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 ${\text{deg}}_H(v)\equiv 1(\mathrm{mod}k)$ 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.

4 pages, comments welcome!