A note on large induced subgraphs with prescribed residues in bipartite graphs

TL;DR

This paper confirms Scott's conjecture that in bipartite graphs without isolated vertices, large induced subgraphs with degrees congruent to 1 modulo k exist with size proportional to the entire graph, specifically proportional to 1/k.

Contribution

The paper proves Scott's conjecture, establishing that the constant c(k) can be taken as proportional to 1/k, which was previously conjectured.

Findings

Confirmed Scott's conjecture on induced subgraphs with prescribed degree residues.

Established that the size of such subgraphs is at least proportional to 1/k of the total graph.

Provided a tight bound up to a constant factor.

Abstract

It was proved by Scott that for every $k\ge2$, 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 $\textrm{deg}_H(v) \equiv 1\pmod{k}$ 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.