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arXiv 1612.08136math.CO
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色顶点Folkman数

Chromatic Vertex Folkman Numbers

Xiaodong Xu, Meilian Liang, Stanisław Radziszowski

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摘要：

对于图$G$和整数$a_1 \ge \cdots \ge a_r \ge 2$，我们记为$G \rightarrow (a_1 ,\cdots ,a_r)^v$当且仅当对于顶点集$V(G)$的每一个$r$ -着色，对于某种颜色$i \in \{1, \cdots, r\}$，在$G$中存在一个单色$K_{a_i}$.顶点Folkman数$F_v(a_1 ,\cdots ,a_r; s)$定义为最小整数$n$，对于该整数，存在一个$n$阶的无$K_s$图$G$，使得$G \rightarrow (a_1 ,\cdots ,a_r)^v$ .众所周知，如果$G \rightarrow (a_1 ,\cdots ,a_r)^v$，则$χ(G) \geq m$，其中$m = 1+ \sum_{i=1}^r (a_i - 1)$。本文研究了具有色数$χ(G)=m$的一类Folkman图$G$，从而引出了色Folkman数的新概念。我们建设性地证明了一些存在性结果，其中包括：对于所有的$r,s \ge 2$，存在$K_{s+1}$ -free图$G$，使得$G \rightarrow (s,\cdots_r,s)^v$和$G$对于这个$r$ -color arrowing成立具有最小可能的色数$r(s-1)+1$.我们还猜想，在某些情况下，我们的构造是最好的，特别是对于每个$s \ge 2$，存在一个$F_v(s,s; s+1)$顶点上的无$K_{s+1}$图$G$，其$χ(G)=2s-1$使得$G \rightarrow (s,s)^v$ .

英文摘要：

For graph $G$ and integers $a_1 \ge \cdots \ge a_r \ge 2$, we write $G \rightarrow (a_1 ,\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \in \{1, \cdots, r\}$. The vertex Folkman number $F_v(a_1 ,\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \rightarrow (a_1 ,\cdots ,a_r)^v$. It is well known that if $G \rightarrow (a_1 ,\cdots ,a_r)^v$ then $χ(G) \geq m$, where $m = 1+ \sum_{i=1}^r (a_i - 1)$. In this paper we study such Folkman graphs $G$ with chromatic number $χ(G)=m$, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all $r,s \ge 2$ there exist $K_{s+1}$-free graphs $G$ such that $G \rightarrow (s,\cdots_r,s)^v$ and $G$ has the smallest possible chromatic number $r(s-1)+1$ for this $r$-color arrowing to hold. We also conjecture that, in some cases, our construction is the best possible, in particular that for every $s \ge 2$ there exists a $K_{s+1}$-free graph $G$ on $F_v(s,s; s+1)$ vertices with $χ(G)=2s-1$ such that $G \rightarrow (s,s)^v$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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