List colorings of $k$-partite $k$-graphs
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909848581439488 |
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| author | Dhawan, Abhishek |
| author_facet | Dhawan, Abhishek |
| contents | 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 $|L(v)| \geq \left((k-1+\varepsilon)Δ/\log Δ\right)^{1/(k-1)}$, then $H$ admits a proper $L$-coloring, provided $Δ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03111 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | List colorings of $k$-partite $k$-graphs Dhawan, Abhishek Combinatorics 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 $|L(v)| \geq \left((k-1+\varepsilon)Δ/\log Δ\right)^{1/(k-1)}$, then $H$ admits a proper $L$-coloring, provided $Δ$ 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. |
| title | List colorings of $k$-partite $k$-graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.03111 |