Degrees in oriented hypergraphs and Ramsey p-chromatic number

TL;DR

This paper generalizes the concept of degree constraints in oriented graphs to hypergraphs, exploring their chromatic properties and connections with hypergraph Ramsey numbers.

Contribution

It introduces a broad generalization of degree-based graph families into hypergraphs and links these to hypergraph Ramsey theory.

Findings

Established a hypergraph degree generalization of $D(k,m)$.

Connected hypergraph degree conditions with Ramsey numbers.

Provided insights into hypergraph coloring under degree constraints.

Abstract

The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $\deg^+(v) \le k$ or (indegree) $\deg^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, into the context of $r$-uniform hypergraphs, using a generalization of Hakimi's theorem to $r$-uniform hypergraphs and by showing some tight connections with the well known Ramsey numbers for hypergraphs.