Determine the optimal universal constant for large odd induced subgraphs

Determine the largest constant c such that every n-vertex graph without isolated vertices contains an odd induced subgraph on at least cn vertices, and in particular determine whether every such graph has an odd induced subgraph of order at least 2n/7.

Background

The paper studies the extremal constant c defined as the infimum of f_o(G)/|V(G)| over all graphs G without isolated vertices, where f_o(G) is the maximum order of an induced odd subgraph of G. Ferber and Krivelevich proved that this constant is positive, with c at least 10{-4}, while Caro observed that c is at most 2/7.

The paper establishes that 2/7 is the exact bound for graphs of maximum degree at most four, but this does not resolve whether 2/7 is the optimal constant for unrestricted graphs. The authors explicitly ask whether graphs exist whose odd induced subgraphs all have order less than 2n/7.

References

This raises a natural question: what is the largest value of $c?$ (Formally, if $f_o(G)$ denotes the size of a largest induced odd subgraph in $G$, then we can define $c$ as the infimum of $f_o(G)/|V(G)|$ over all graphs $G$ without isolated vertices.) Caro mentioned, as a known fact, that $c\le \frac{2}{7}.$ Are there $n$-vertex graphs without isolated vertices in which all odd induced subgraphs have less than $2n/7$ vertices?

— Odd Induced Subgraphs in Graphs of Maximum Degree Four  (2511.15489 - Ai et al., 19 Nov 2025) in Section 1, Introduction; Abstract