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Generalized subdivisions in digraphs spanned by subdivision of smaller digraphs and the chromatic number | ||
Communications in Combinatorics and Optimization | ||
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 16 خرداد 1404 اصل مقاله (409.21 K) | ||
نوع مقاله: Original paper | ||
شناسه دیجیتال (DOI): 10.22049/cco.2025.29422.1988 | ||
نویسنده | ||
Salman Ghazal* | ||
College of Engineering and Technology, American University of the Middle East, Egaila, 54200, Kuwait | ||
چکیده | ||
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. | ||
کلیدواژهها | ||
Oriented cycle؛ Hamiltonian؛ Chromatic number؛ Subdivision | ||
مراجع | ||
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[8] R. Kim, S.J. Kim, J. Ma, and B. Park, Cycles with two blocks in k-chromatic digraphs, J. Graph Theory 88 (2018), no. 4, 592–605. https://doi.org/10.1002/jgt.22232
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