论文标题
在平衡的上色数量的有限射击飞机上
On the balanced upper chromatic number of finite projective planes
论文作者
论文摘要
在本文中，我们研究了超图的顶点着色，其中所有颜色级别的大小最多都不同于一种（平衡的着色），并且每个颜色的大小都具有至少两个相同颜色的顶点（无彩虹色的着色）。对于任何HyperGraph $ h $，最大数字$ k $，其中有一个平衡的无彩虹$ k $ - 颜色为$ h $，称为平衡的超图的上色数。我们通过确定所有$ q $的desarguesian投射平面$ \ mathrm {pg} $ $ \ mathrm {pg}（2，q）$来确定Araujo-Pardo，Kiss和Montejano的猜想。此外，我们渐近地确定了非降低投影平面的几个家族的平衡上色数，并使用概率方法为任意投影平面提供了一般的下限，该概率方法将参数确定为乘法常数。
In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $H$, the maximum number $k$ for which there is a balanced rainbow-free $k$-coloring of $H$ is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane $\mathrm{PG}(2,q)$ for all $q$. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.
