Balanced independent sets and colorings of hypergraphs
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929436054519808 |
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| author | Dhawan, Abhishek |
| author_facet | Dhawan, Abhishek |
| contents | A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I\cap V_j|$ for each $1 \leq i, j \leq k$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $α_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $χ_b(H)$ in terms of the maximum degree $Δ$. Our results recover those of recent work of Chakraborti for $k = 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01940 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Balanced independent sets and colorings of hypergraphs Dhawan, Abhishek Combinatorics Discrete Mathematics A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I\cap V_j|$ for each $1 \leq i, j \leq k$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $α_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $χ_b(H)$ in terms of the maximum degree $Δ$. Our results recover those of recent work of Chakraborti for $k = 2$. |
| title | Balanced independent sets and colorings of hypergraphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2311.01940 |