AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.
Hadwiger's conjecture (1943) states that every graph $G$ with chromatic number at least $t$ contains a $K_t$-model: a collection of $t$ vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that for all $1\le i<j\le t$ some vertex in $T_j$ has a neighbour in $T_i$. Replacing"some"in this definition by"every"gives ri...
The well-known Bermond-Thomassen conjecture states that every digraph of minimum out-degree at least $2k-1$ contains $k$ vertex-disjoint directed cycles. Despite being posed in 1981, this conjecture remains unresolved for all $k \ge 4$. We prove a relaxation of this conjecture: every digraph $D$ of minimum out-degree a...
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