• DOI:10.1002/jgt.22232
• Corpus ID: 67772744

Cycles with two blocks in k‐chromatic digraphs

@article{Kim2016CyclesWT,
  title={Cycles with two blocks in k‐chromatic digraphs},
  author={Ringi Kim and Seog-Jin Kim and Jie Ma and Boram Park},
  journal={Journal of Graph Theory},
  year={2016},
  volume={88},
  pages={592 - 605}
}

• Ringi Kim, Seog-Jin Kim, +1 author Boram Park
• Published 19 October 2016
• Mathematics
• Journal of Graph Theory

Let k and ℓ be positive integers. A cycle with two blocks c(k,ℓ) is a digraph obtained by an orientation of an undirected cycle, which consists of two internally (vertex) disjoint paths of lengths at least k and ℓ, respectively, from a vertex to another one. A problem of Addario‐Berry, Havet and Thomassé [J. Combin. Theory Ser. B 97 (2007), 620–626] asked if, given positive integers k and ℓ such that k+ℓ≥4 , any strongly connected digraph D containing no c(k,ℓ) has chromatic number at most k+ℓ−1 . In this article, we show that such digraph D has chromatic number at most O((k+ℓ)2) , improving the previous upper bound O((k+ℓ)4) of Cohen et al. [Subdivisions of oriented cycles in digraphs with large chromatic number, to appear]. We also show that if in addition D is Hamiltonian, then its underlying simple graph is (k+ℓ−1) ‐degenerate and thus the chromatic number of D is at most k+ℓ , which is tight.