AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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A digraph is antidirected if every vertex has indegree zero or outdegree zero. Let $k\ge2$, and let $F$ be an antidirected forest with $k$ arcs and no isolated vertices. We prove that every digraph $D$ of order $n$ with more than $g_k(n):=2\max\left\{\binom{2k-1}{2}, (k-1)\left(n-\frac{k}{2}\right)\right\}$ arcs contai...
Let $\Cvec$ be a fixed orientation of the cycle $C_\ell$, $\ell\ge3$, which is not directed. For an oriented graph $D$, let $d_D^*(v):=\max\{d_D^+(v),d_D^-(v)\},$ and let \[ \sigore(D):=\min\bigl\{d_D^*(x)+d_D^*(y):x\ne y,\ xy,yx\notin A(D)\bigr\}, \] with $\sigore(D)=\infty$ if the underlying graph of $D$ is complete....
Let $\overrightarrow{C}$ be any orientation of the cycle $C_{\ell}$ which is not directed. We prove that, for every integer $\ell\ge3$ and every $\mu>0$, there is a real $\gamma$ such that every sufficiently large oriented graph $D$ with $\ell\mid |D|$, minimum semidegree at least $(1/4+\mu)|D|$ and independence number...
Let $C_{2s}^{\mathrm{ad}}$ be the anti-directed cycle of length $2s$, where $s\geq2$. We prove that, for every $\mu>0$, every sufficiently large $n$-vertex oriented graph $D$ with $2s\mid n$, \[ \delta^0(D)\geq\left(\frac14+\mu\right)n \qquad\text{and}\qquad \alpha(D)=o(n) \] contains a $C_{2s}^{\mathrm{ad}}$-factor. T...
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