Determine the optimal constant for maximum-degree-five graphs

Determine the largest constant c such that every n-vertex graph without isolated vertices and with maximum degree at most five contains an odd induced subgraph on at least cn vertices, and establish whether this constant is at least 2/7.

Background

The paper proves that the optimal constant is 2/7 for graphs with maximum degree at most four, extending the previously known sharp value 2/5 for graphs with maximum degree at most three.

The next degree case is left unresolved. Establishing a lower bound of 2/7 for maximum-degree-five graphs would provide additional evidence that 2/7 may be the optimal constant for all graphs.

References

What is the largest value of $c$ for graphs with maximum degree five? If it is no less than $\tfrac{2}{7}$ then it would further support the possibility that $\tfrac{2}{7}$ is the optimal constant for all graphs.

— Odd Induced Subgraphs in Graphs of Maximum Degree Four  (2511.15489 - Ai et al., 19 Nov 2025) in Section 3, Conclusion