论文标题
将图形分配到不包含规定集团的诱导子图中
Partitioning of a graph into induced subgraphs not containing prescribed cliques
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Let $K_p$ be a complete graph of order $p\geq 2$. A $K_p$-free $k$-coloring of a graph $H$ is a partition of $V(H)$ into $V_1, V_2\ldots,V_k$ such that $H[V_i]$ does not contain $K_p$ for each $i\leq k $. In 1977 Borodin and Kostochka conjectured that any graph $H$ with maximum degree $Δ(H)\geq 9$ and without $K_{Δ(H)}$ as a subgraph has chromatic number at most $Δ(H)-1$. As analogue of the Borodin-Kostochka conjecture, we prove that if $p_1\geq \cdots\geq p_k\geq 2$, $p_1+p_2\geq 7$, $\sum_{i=1}^kp_i=Δ(H)-1+k$, and $H$ does not contain $K_{Δ(H)}$ as a subgraph, then there is a partition of $V(H)$ into $V_1,\ldots,V_k$ such that for each $i$, $H[V_i]$ does not contain $K_{p_i}$. In particular, if $p\geq 4$ and $H$ does not contain $K_{Δ(H)}$ as a subgraph, then $H$ admits a $K_p$-free $\lceil{Δ(H)-1\over p-1}\rceil$-coloring. Catlin showed that every connected non-complete graph $H$ with $Δ(H)\geq 3$ has a $Δ(H)$-coloring such that one of the color classes is maximum $K_2$-free subset (maximum independent set). In this regard, we show that there is a partition of vertices of $H$ into $V_1$ and $V_2$ such that $H[V_1]$ does not contain $K_{p}$, $H[V_2]$ does not contain $K_{q}$, and $V_1$ is a maximum $K_p$-free subset of V(H) if $p\geq 4$, $q\geq 3$, $p+q=Δ(H)+1$, and its clique number $ω(H)=p$.
