List colorings of \(k\)-partite \(k\)-graphs

Abhishek Dhawan¹

¹Georgia Institute of Technology

The electronic journal of combinatorics, Tome 32 (2025) no. 2

Résumé

A $k$-uniform hypergraph (or $k$-graph) $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that each edge in $E$ contains precisely one vertex from each $V_i$. In this note, we consider list colorings for such hypergraphs. We show that for any $\varepsilon > 0$ if each vertex $v \in V(H)$ is assigned a list of size $|L(v)| \geq \left((k-1+\varepsilon)\Delta/\log \Delta\right)^{1/(k-1)}$, then $H$ admits a proper $L$-coloring, provided $\Delta$ is sufficiently large. Up to a constant factor, this matches the bound on the chromatic number of simple $k$-graphs shown by Frieze and Mubayi, and that on the list chromatic number of triangle free $k$-graphs shown by Li and Postle. Our results hold in the more general setting of "color-degree" as has been considered for graphs. Furthermore, we establish a number of asymmetric statements matching results of Alon, Cambie, and Kang for bipartite graphs.

Détail

DOI : 10.37236/12657

Classification : 05C15, 05C35, 05C65, 05D40
Mots-clés : \(k\)-uniform hypergraph, list chromatic number of triangle free \(k\)-graphs