Dichromatic number and forced subdivisions

Lior Gishboliner, Raphael Steiner, Tibor Szabó

    Research output: Contribution to journalArticlepeer-review

    Abstract

    We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph F, denote by maderχ→(F) the smallest integer k such that every k-dichromatic digraph contains a subdivision of F. As our first main result, we prove that if F is an orientation of a cycle then maderχ→(F)=v(F). This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is that maderχ→(F)=4 for every tournament F of order 4. This is an extension of the classical result by Dirac that 4-chromatic graphs contain a K4-subdivision to directed graphs.

    Original languageEnglish
    Pages (from-to)1-30
    Number of pages30
    JournalJournal of Combinatorial Theory. Series B
    Volume153
    DOIs
    StatePublished - Mar 2022

    Keywords

    • Dichromatic number
    • Digraph
    • Subdivision
    • Topological minor

    Fingerprint

    Dive into the research topics of 'Dichromatic number and forced subdivisions'. Together they form a unique fingerprint.

    Cite this