# Subdivisions of oriented cycles in digraphs with large chromatic number

@article{Cohen2016SubdivisionsOO,
title={Subdivisions of oriented cycles in digraphs with large chromatic number},
author={Nathann Cohen and Fr{\'e}d{\'e}ric Havet and William Lochet and Nicolas Nisse},
journal={Journal of Graph Theory},
year={2016},
volume={89},
pages={439 - 456}
}
• Published 25 May 2016
• Mathematics
• Journal of Graph Theory
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out‐degree 2 and two…
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A four blocks cycle C(k1, k2, k3, k4) is an oriented cycle formed by the union of four internally disjoint directed paths of lengths k1, k2, k3, and k4 respectively. El Mniny[2] proved that if D is a
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The (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle is studied, which generalize both the Maximum Flow and Longest Path problems.
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It is conjectured that for any positive integers $k_1, k_2,k_3$, there is an integer g(k-1,k-2, k-3) such that every strongly connected digraph with chromatic number greater than g contains a subdivision of $B(k_2; k_3)$.

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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 L(G)={5,7} implies χ(G)=3.5, and Wang completely analyzed the case, and Camacho proved that if L( G)={k,k+2}, k≥5, then χ (G)≤4.
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