Coloring directed hypergraphs
Bal\'azs Keszegh

TL;DR
This paper explores coloring properties of directed hypergraphs, proving a conjecture for 3-uniform cases, and shows that certain intersection restrictions guarantee a 2-colorability, advancing hypergraph coloring theory.
Contribution
It proves a conjecture on 2-colorability of directed hypergraphs with intersection restrictions specifically for 3-uniform hypergraphs, extending prior extremal results.
Findings
Proved the conjecture for 3-uniform directed hypergraphs.
Established that avoiding a specific hypergraph structure ensures 2-colorability.
Extended understanding of coloring in directed hypergraphs beyond extremal edge counts.
Abstract
Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertices of each hyperedge is partitioned into two parts, a tail and a head. We present a conjecture of D. P\'alv\"olgyi and the author, which states that directed hypergraphs with a certain restriction on their pairwise intersections can be colored with two colors. Besides other contributions, our main result is a proof of this conjecture for -uniform directed hypergraphs. This result can be phrased equivalently such that if a -uniform directed hypergraph avoids a certain directed hypergraph with two hyperedges, then it admits a proper -coloring. Previously, only extremal problems regarding the maximum number of edges of directed hypergraphs that avoid a certain hyperedge were studied.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory
