The spectral radius of $k$-chromatic $r$-graphs

Abstract

For an $r$-uniform hypergraph $G$, let ${\lambda }^{(p)}(G)$ denote its $p$-spectral radius, defined as the maximum of the polyform of $G$ over the unit sphere in the ${\ell }_p$-norm. Let $Q_k^r(n)$ be the complete $k$-chromatic $r$-graph on $n$ vertices with color classes as equal as possible. Kang--Nikiforov--Yuan conjectured that, for every $p\ge 1$ and $n>(r-1)k$, the $r$-graph $Q_k^r(n)$ is the unique maximizer of ${\lambda }^{(p)}$ among all $k$-chromatic $r$-graphs of order $n$. They also conjectured the corresponding explicit bound \[ λ^{(p)}(G) \le r!\left(\tbinom nr-k\tbinom{n/k}{r}\right)n^{-r/p}, \] with equality only in the divisible extremal case. The case $r=3$ was established in their work. This paper resolves the remaining cases $r\ge 4$, and hence settles both conjectures for all $r\ge 3$. As a consequence, the same threshold gives an anti-Wilf-type spectral certificate: any $r$-graph of order $n$ whose $p$-spectral radius exceeds the displayed bound has chromatic number at least $k+1$.