# Bispindle in strongly connected digraphs with large chromatic number

@article{Cohen2017BispindleIS,
title={Bispindle in strongly connected digraphs with large chromatic number},
author={Nathann Cohen and Fr{\'e}d{\'e}ric Havet and William Lochet and Raul Lopes},
journal={Electron. Notes Discret. Math.},
year={2017},
volume={62},
pages={69-74}
}
• Published 7 March 2017
• Mathematics
• Electron. Notes Discret. Math.
2 Citations
• Mathematics
• 2020
A $(2+1)$-bispindle $B(k_1,k_2;k_3)$ is the union of two $xy$-dipaths of respective lengths $k_1$ and $k_2$, and one $yx$-dipath of length $k_3$, all these dipaths being pairwise internally disjoint.
Le but principal de cette these est de presenter des conditions suffisantes pour garantir l'existence de subdivisions dans les graphes diriges. Bien que ce genre de questions soit assez bien maitrise

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If $D$ is an oriented path, or an in-arborescence, or the union of two directed paths from x to y and a directed path from y tox, then every digraph with minimum out-degree large enough contains a subdivision of $D$.
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This paper is able to find a digraph which shows that the answer to the above problem is no and it is shown that if in addition £D is Hamiltonian, then its underlying simple graph is $(k+\ell-1)$-degenerate and thus the chromatic number of $D$ is at most $k-\ell$, which is tight.
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It is shown that for any oriented cycle C, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number and for any cycle with two blocks, every strongly connected digraph with sufficiently large Chromatic number contains a subdivision of C.