Treedepth vs circumference

The circumference of a graph $G$ is the length of a longest cycle in $G$, or$+\infty$ if $G$ has no cycle. Birmel\'e (2003) showed that the treewidth of agraph $G$ is at most its circumference minus one. We strengthen this result for$2$-connected graphs as follows: If $G$ is $2$-connected, then its treedepth isat most its circumference. The bound is best possible and improves on anearlier quadratic upper bound due to Marshall and Wood (2015).<br

Separating Polynomial χ -Boundedness from χ -Boundedness

AbstractExtending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $$f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}$$ f : N → N ∪ { ∞ } with $$f(1)=1$$ f ( 1 ) = 1 and $$f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right)$$ f ( n ) ⩾ 3 n + 1 3 , we construct a hereditary class of graphs $${\mathcal {G}}$$ G such that the maximum chromatic number of a graph in $${\mathcal {G}}$$ G with clique number n is equal to f(n) for every $$n\in \mathbb {N}$$ n ∈ N . In particular, we prove that there exist hereditary classes of graphs that are $$\chi$$ χ -bounded but not polynomially $$\chi$$ χ -bounded.</jats:p