论文标题
反van der waerden循环的图形数量
Anti-van der Waerden Numbers of Graph Products of Cycles
论文作者
论文摘要
图$ g $中的$ k $ - term算术进程（$ k $ -ap）是一个顶点列表，使每个连续的顶点相距相同。如果$ c $是$ g $的顶点的着色函数，而$ g $的$ k $ -ap具有明显的颜色，那么$ k $ -ap是彩虹$ k $ -ap。相对于$ k $的图形$ g $的反van der der waerden是最不正面的整数$ r $，因此每种带有域$ v（g）$的过滤着色和codomain $ \ {1,2，\ dots，r \}，r \} = [r] = [r] $保证具有彩虹$ k $ -ap。本文重点介绍了带有周期的$ 3 $ paps和图形产品。具体而言，针对$ 3 $的反van der der der waerden号码是精确确定的，$ p_m \ square c_n $，$ c_m \ square c_n $和$ g \ square c_ {2n+1} $。
A $k$-term arithmetic progression ($k$-AP) in a graph $G$ is a list of vertices such that each consecutive pair of vertices is the same distance apart. If $c$ is a coloring function of the vertices of $G$ and a $k$-AP in $G$ has each vertex colored distinctly, then that $k$-AP is a rainbow $k$-AP. The anti-van der Waerden number of a graph $G$ with respect to $k$ is the least positive integer $r$ such that every surjective coloring with domain $V(G)$ and codomain $\{1,2,\dots,r\} = [r]$ is guaranteed to have a rainbow $k$-AP. This paper focuses on $3$-APs and graph products with cycles. Specifically, the anti-van der Waerden number with respect to $3$ is determined precisely for $P_m \square C_n$, $C_m\square C_n$ and $G\square C_{2n+1}$.
