Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 06 06 Generalized subdivisions in digraphs spanned by subdivision of smaller digraphs and the chromatic number 14962 10.22049/cco.2025.29422.1988 EN Salman Ghazal College of Engineering and Technology, American University of the Middle East, Egaila, 54200, Kuwait Journal Article 2024 02 01 A generalized subdivision $H'$ of a digraph $H$ is obtained by replacing each arc $e=(x,y)\in E(H)$ with tail $x$ and head $y$, by an oriented path $P_e$ whose first arc has tail $x$ and whose last arc has head $y$, all these new paths being internally disjoint. If all these new paths are directed ones, then $H'$ is simply a subdivision of $H$. The number of blocks (which turns out to have the same parity of $|E(H)|$) of the generalized subdivision $H'$ of $H$ is the sum of all the number of blocks of the new paths $P_e$. In this paper, we prove that if $D$ is spanned by a subdivision of a digraph $H$ such that $\chi(D)$ is at least $2n+|V(H)|+|E(H)|$, then $D$ contains a generalized subdivision of $H$ with $n$ blocks. This bound is simplified when $H$ is an oriented tree. If $H$ is an oriented cycle, then our results assert a special case of a conjecture of Cohen et al. Moreover, the bound is improved to $2n+1$ if $H$ is an oriented cycle with two blocks or $H$ is a directed cycle. https://comb-opt.azaruniv.ac.ir/article_14962_e3f9199426f7b4d0c8fdd340a68ae8cb.pdf