排除路径和双团
Excluding paths and bicliques
AI总结：
研究排除路径和双团作为诱导子图的图类，改进了界定路径最大长度函数\(f\)的界，证明其为关于\(\omega^c\)的单指数函数且最优，还得出此类图树深度由团数多项式函数界定及相关遗传图类结论。
AI中文摘要：
在文献中，广泛研究了排除路径和双团作为诱导子图的图类。此类图的关键结构结果之一是加尔文、里瓦尔和桑兹（1982年）给出的一个拉姆齐型结果，该结果确定了一个函数\(f\)，它根据团数\(\omega\)界定路径的最大长度。我们将\(f\)的已知最佳界改进为一个关于\(\omega^c\)的单指数函数，其中\(c\)为某个常数，我们证明这在优化\(c\)的情况下是最优的。我们的方法对树深度也有影响。特别地，我们表明，对于排除路径和双团作为诱导子图的图，树深度由团数的多项式函数界定。反过来，这个结果意味着每个允许根据团数界定图类中树深度的函数的遗传图类，都允许一个多项式这样的函数。这给出了哈杰比（2025年）关于路径宽度的一个近期结果的树深度类似物。
英文摘要：
Classes of graphs excluding a path and a biclique as induced subgraphs are extensively studied in the literature. One of the key structural results for such graphs is a Ramsey-type result due to Galvin, Rival, and Sands (1982), establishing the existence of a function $f$ bounding the maximum length of a path in terms of clique number $ω$. We improve the best known bound on $f$ to a function that is a singly exponential in $ω^c$, for some constant $c$, which we show is best possible, up to optimizing $c$. Our approach also has consequences for treedepth. In particular, we show that, for graphs excluding a path and a biclique as induced subgraphs, treedepth is bounded by a polynomial function of clique number. In turn, this result implies that every hereditary graph class that admits a function bounding treedepth of graphs in the class in terms of clique number, admits a polynomial such function. This gives a treedepth analogue of a recent result on pathwidth due to Hajebi (2025).
