Counterexamples to a conjecture and improved bounds for bipanchromatic colorings

Overview & Synopsis

We study the bipanchromatic number $χ_B$, introduced by Lalou et al. (Discrete Appl. Math. 391 (2026), 511-522) as the maximum number of colors in a panchromatic hypergraph coloring where every color appears at least twice. They show that merging uniquely occurring colors in a maximum panchromatic coloring gives the lower bound $β=χ_P-\lceilα/2\rceil \leq χ_B$, where $χ_P$ is the panchromatic number and $α$ is the minimum number of such colors among maximum panchromatic colorings. They conjectured that $χ_B=β$ and subsequently disproved it (arXiv:2610.08042). We identify the smallest counterexample, with six vertices and three hyperedges, both minimum possible. We show that the ratio bound $χ_B/β