Sarah Frederickson | Searcharxiv

arXiv2026

Improved bounds for the chromatic index of $k$-uniform hypergraphs

In 1997, Alon and Kim conjectured that if $H$ is a $k$-uniform $t$-simple hypergraph with maximum degree $D$ sufficiently large, then the chromatic index $\chi'(H)$ is upper bounded by $(t-1+1/t+\varepsilon)D$. Using probabilistic techniques and a nibble coloring method, we prove a general coloring theorem stating that a $k$-uniform $t$-simple hypergraph $H$ with large maximum degree $D$ satisfies $$\chi'(H) \le (b+\varepsilon)kD,$$ where $b$ is a particular parameter derived from local structural information about $H$. We use structural techniques to prove sharp upper bounds on $b$ in the 3-uniform 2-simple, and 3-uniform 3-simple cases. In particular, we deduce as a corollary that for sufficiently large $D$, every 3-uniform 2-simple and 3-simple hypergraph of maximum degree at most $D$ has chromatic index at most $2.3581D$ and $2.6791D$, respectively.

math.CO↗

arXiv2026

Almost all graphs are vertex-minor universal

Answering a question of Claudet, we prove that the uniformly random graph $G\sim \mathbb G(n, 1/2)$ is $\Omega(\sqrt n)$-vertex-minor universal with high probability. That is, for some constant $\alpha\approx 0.911$, any graph on any $\alpha\sqrt n$ specified vertices of $G$ can be obtained as a vertex-minor of $G$. This has direct implications for quantum communications networks: an $n$-vertex $k$-vertex-minor universal graph corresponds to an $n$-qubit $k$-stabilizer universal graph state, which has the property that one can induce any stabilizer state on any $k$ qubits using only local operations and classical communications. We further employ our methods in two other contexts. We obtain a bipartite pivot-minor version of our main result, and we use it to derive a universality statement for minors in random binary matroids. We also introduce the vertex-minor Ramsey number $R_{\mathrm{vm}}(k)$ to be the smallest value $n$ such that every $n$-vertex graph contains an independent set of size $k$ as a vertex-minor. Supported by our main result, we conjecture that $R_{\mathrm{vm}}(k)$ is polynomial in $k$. We prove $\Omega(k^2) \leq R_{\mathrm{vm}}(k) \leq 2^k - 1$.

quant-ph↗