A hereditary class of graphs $\mathcal{G}$ is χ-bounded if there exists a function f such that every graph $G\in \mathcal{G}$ satisfies $\chi (G)⩽f(\omega (G))$, where $\chi (G)$ and $\omega (G)$ 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 $\chi (G)⩽{(t-1)}^{\omega (G)-1}$. In 2017, Chudnovsky, Scott, and Seymour proved that $C_{⩾t}$-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 $C_{⩾t}$-free. It remains a major open problem in the area whether for $C_{⩾t}$-free, or at least $P_t$-free graphs G, the value of $\chi (G)$ can be bounded from above by a polynomial function of $\omega (G)$. 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 every ℓ and every $C_{⩾t}$-free graph which does not contain $K_{\ell ,\ell }$ as a subgraph, it holds that $\chi (G)⩽{\ell }^c$.

图的遗传类 $\mathcal{G}$是χ 有界的，如果存在一个函数f使得每个图$G\in \mathcal{G}$ 满足 $\chi (G)⩽F(\omega (G))$， 在哪里 $\chi (G)$ 和 $\omega (G)$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，如果G是${磷}_{吨}$-free，即不包含t顶点路径作为诱导子图，则$\chi (G)⩽{(吨-1)}^{\omega (G)-1}$. 2017 年，Chudnovsky、Scott 和 Seymour 证明了$C_{⩾吨}$-free 图，即排除至少具有t个顶点的诱导循环的图，也是χ -有界的，并且获得的界限再次是团数中的超多项式。注意${磷}_{吨-1}$-free 图尤其是 $C_{⩾吨}$-自由。在该地区仍然是一个重大的开放问题，无论是$C_{⩾吨}$- 免费，或至少 ${磷}_{吨}$- 自由图G，值$\chi (G)$ 可以从上面的多项式函数 $\omega (G)$. 我们考虑放松这个问题，我们将色数与图中包含的最大平衡双角的大小进行比较，作为（不一定是诱导）子图。我们证明对于每个t 都存在一个常数c使得对于每个ℓ和每个$C_{⩾吨}$- 不包含的自由图 ${钾}_{\ell ,\ell }$ 作为子图，它认为 $\chi (G)⩽{\ell }^C$.