$t$ -无扫帚图的多项式$χ$ -结合函数
Polynomial $χ$-binding functions for $t$-broom-free graphs
摘要：
对于任何正整数$t$，\emph{$t$-broom}是通过将边细分一次而从$K_{1,t+1}$获得的图。本文证明了：对于没有导出$t$ -簇的图$G$，我们有$χ(G) = o(ω(G)^{t+1})$，其中$χ(G)$和$ω(G)$分别是$G$的色数和团数.当$t=2$，这回答了Schiermeyer和Randerath的问题。此外，对于$t=2$，我们将$χ(G)$上的界加强到$7ω(G)^2$，证实了Sivaraman的一个猜想。对于$t\geq 3$和\{ $t$ -broom，$K_{t,t}$ \}-free图，我们将其界改进为$o(ω^{t})$ .
英文摘要：
For any positive integer $t$, a \emph{$t$-broom} is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $χ(G) = o(ω(G)^{t+1})$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. When $t=2$, this answers a question of Schiermeyer and Randerath. Moreover, for $t=2$, we strengthen the bound on $χ(G)$ to $7ω(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and \{$t$-broom, $K_{t,t}$\}-free graphs, we improve the bound to $o(ω^{t})$.
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。
