关于二部多重图的Ramsey数
On the Ramsey Numbers for Bipartite Multigraphs
摘要：
完全二部图的着色是混洗保持的，如果是这样的情况，即向边$(u, v)$和$(u', v')$分配颜色$c$强制对边$(u, v')$和$(u',v)$进行相同的颜色分配。(In换句话说，关于颜色$c$的导出子图是完全的。）在本文中，我们研究一类完全二部多重图的Ramsey问题的一个变种。(By多重图是指允许有多条边但不允许有环的图。与Ramsey理论中的传统m-着色方案不同，该方案施加了一个约束（即，$m$）的总数量的颜色允许在一个图中，我们引入了一个宽松的版本称为m-局部着色，它只需要，对于每个顶点$v$，与$v$的事件边的颜色的数量是有界的$m$。请注意，$m$ -局部着色下的图中找到的颜色数量可能超过m。我们证明了，给定任何$n \times n$完全二部多重图$G$，每个洗牌保持$m$ -局部着色显示$K_{p,p}$的单色副本，只要$2(p-1)(m-1) < n$。此外，当（i）$m=2$，或（ii）$n=2^k$和$m=3\cdot 2^{k-2}$对于每个整数$k\geq 2$时，上述界是紧的。对于$p$的下界，我们证明了如果$p> \lceil \frac{n}{m} \rceil$，则不能保证单色$K_{p,p}$的存在。最后，我们给出了$k$ -部图的一个推广和一个适用于一般图的方法. $m$ -局部着色中的许多结论可以推广到$m$ -着色中类似的结果.
英文摘要：
A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph with respect to color $c$ is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs. (By a multigraph we mean a graph in which multiple edges, but no loops, are allowed.) Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., $m$) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring which only requires that, for every vertex $v$, the number of colors associated with $v$'s incident edges is bounded by $m$. Note that the number of colors found in a graph under $m$-local coloring may exceed m. We prove that given any $n \times n$ complete bipartite multigraph $G$, every shuffle-preserved $m$-local coloring displays a monochromatic copy of $K_{p,p}$ provided that $2(p-1)(m-1) < n$. Moreover, the above bound is tight when (i) $m=2$, or (ii) $n=2^k$ and $m=3\cdot 2^{k-2}$ for every integer $k\geq 2$. As for the lower bound of $p$, we show that the existence of a monochromatic $K_{p,p}$ is not guaranteed if $p> \lceil \frac{n}{m} \rceil$. Finally, we give a generalization for $k$-partite graphs and a method applicable to general graphs. Many conclusions found in $m$-local coloring can be inferred to similar results of $m$-coloring.
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