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

TL;DR

This paper proves conjectures about the maximum $p$-spectral radius of $k$-chromatic $r$-graphs, confirming the extremal structures and bounds for all $r \\ge 3$, with implications for chromatic number thresholds.

Contribution

It resolves the remaining cases of conjectures on the spectral radius of $k$-chromatic $r$-graphs for all $r \\ge 4$, confirming the extremal configurations and bounds.

Findings

Confirmed the conjectured maximum $p$-spectral radius for all $r \\ge 3$.

Identified the extremal $k$-chromatic $r$-graphs achieving the maximum.

Established spectral thresholds for chromatic number based on the $p$-spectral radius.

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 $p\ge1$ 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 \[ \lambda^{(p)}(G) \le r!\left(\tbinom nr-k\tbinom{n/k}{r}\right)n^{-r/p}, \] with equality only in the divisible extremal case. The case $r=3$ was established in their work. This paper resolves the remaining cases $r\ge4$, and hence settles both conjectures for all $r\ge3$. 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 $k+1$.