arXiv 2012.036862020-12math.CO

不含大完全二部子图的$P_t$ -free和$C_{\geq t}$ -free图的退化

Degeneracy of $P_t$-free and $C_{\geq t}$-free graphs with no large complete bipartite subgraphs

Marthe Bonamy, Nicolas Bousquet, Michał Pilipczuk, Paweł Rzążewski, Stéphan Thomassé, Bartosz Walczak

摘要：

图的遗传类$\mathcal{G}$称为\emph{$χ$-bounded}，如果存在函数$f$使得每个图$G \in \mathcal{G}$满足$χ(G) \leq f(ω(G))$，其中$χ(G)$和$ω(G)$分别是$G$的色数和团数.作为关于$χ$ -有界类的第一个结果之一，Gyárfás在1985年证明了，如果$G$是$P_t$ -free的，即，不包含$t$ -顶点路径作为导出子图，则$χ(G) \leq (t-1)^{ω(G)-1}$。2017年，Chudnovsky，Scott和Seymour证明了无$C_{\geq t}$图，即，图，排除诱导圈至少$t$顶点，是$χ$有界的，以及，并得到的界再次超多项式的团数。注意，无$P_{t-1}$的图特别是无$C_{\geq t}$的.它仍然是一个主要的开放问题，在该地区是否为$C_{\geq t}$ -自由，或至少$P_t$ -自由图$G$，$χ(G)$的值可以有界从上由一个多项式函数的$ω(G)$。我们考虑这个问题的一个放松，在那里我们比较色数的大小最大的平衡biclique包含在图中作为一个（不一定诱导）子图。我们证明了对每个$t$都存在一个常数$c$，使得对和每个不含$K_{\ell,\ell}$作为子图的$C_{\geq t}$ -free图都成立：$χ(G) \leq \ell^{c}$ .

英文摘要：

A hereditary class of graphs $\mathcal{G}$ is \emph{$χ$-bounded} if there exists a function $f$ such that every graph $G \in \mathcal{G}$ satisfies $χ(G) \leq f(ω(G))$, where $χ(G)$ and $ω(G)$ are the chromatic number and the clique number of $G$, respectively. As one of the first results about $χ$-bounded classes, Gyárfás proved in 1985 that if $G$ is $P_t$-free, i.e., does not contain a $t$-vertex path as an induced subgraph, then $χ(G) \leq (t-1)^{ω(G)-1}$. In 2017, Chudnovsky, Scott, and Seymour proved that $C_{\geq t}$-free graphs, i.e., graphs that exclude induced cycles with at least $t$ vertices, are $χ$-bounded as well, and the obtained bound is again superpolynomial in the clique number. Note that $P_{t-1}$-free graphs are in particular $C_{\geq t}$-free. It remains a major open problem in the area whether for $C_{\geq t}$-free, or at least $P_t$-free graphs $G$, the value of $χ(G)$ can be bounded from above by a polynomial function of $ω(G)$. We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every $t$ there exists a constant $c$ such that for and every $C_{\geq t}$-free graph which does not contain $K_{\ell,\ell}$ as a subgraph, it holds that $χ(G) \leq \ell^{c}$.

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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