Decomposing and colouring some locally semicomplete digraphs

Heroes in oriented complete multipartite graphs

“…and Δ (1,3,2) are not heroes in oriented complete multipartite graphs. Since reversing all arcs of a C Δ(1, 2, )…”

“…3 nor Δ (1,2,3) are heroes in oriented complete multipartite graphs. This is implied by the existence of C {Δ(1, 2, ), Δ(1, 2, 3)} 3 -free oriented complete multipartite graphs with arbitrarily large dichromatic number.…”

“…Proof. Suppose that D contains a copy of Δ (1,2,2) and name its vertices as in Figure 3. Since Δ(1, 2, 2) is a tournament, v i 's are contained in pairwise distinct parts of D, and thus are totally ordered.…”

On coloring digraphs with forbidden induced subgraphs

“…Aboulker et al considered the directed path → P 4 and showed in one of their main results that the set other oriented paths on four vertices. Two of them, which are called P (2,1) + and P (2,1) − in [2], consist of two oppositely oriented dipaths of length two and one, respectively.…”

“…After the submission of this manuscript, independently discovered proofs both for Theorem 1 and Remark 5.2 (which appears in the conclusion) have appeared in the arXiv‐preprint [1] by Aboulker, Aubian, and Charbit. Their proof of Theorem 1 is quite different, as they obtain and use a full structural characterization of the class $${\text{Forb}}_{\text{ind}}({\overleftrightarrow{K}}_{2},{S}_{2}^{+},{W}_{3}^{+})$$ of digraphs.…”