January 2025
·
1 Read
·
2 Citations
A k k ‐uniform hypergraph H = ( V , E ) H = ( V , E ) is k k ‐partite if V V can be partitioned into k k sets V 1 , … , V k V 1 , \unicode{x02026} , V k such that every edge in E E contains precisely one vertex from each V i V i . We call such a graph n n ‐balanced if ∣ V i ∣ = n \unicode{x02223} V i \unicode{x02223} = n for each i i . An independent set I I in H H is balanced if ∣ I ∩ V i ∣ = ∣ I ∩ V j ∣ \unicode{x02223} I \unicode{x02229} V i \unicode{x02223} = \unicode{x02223} I \unicode{x02229} V j \unicode{x02223} for each 1 ⩽ i , j ⩽ k 1 \unicode{x02A7D} i , j \unicode{x02A7D} k , and a coloring is balanced if each color class induces a balanced independent set in H H . In this paper, we provide a lower bound on the balanced independence number α b ( H ) \unicode{x003B1} b ( H ) in terms of the average degree D = ∣ E ∣ / n D = \unicode{x02223} E \unicode{x02223} / n , and an upper bound on the balanced chromatic number χ b ( H ) \unicode{x003C7} b ( H ) in terms of the maximum degree Δ \unicode{x00394} . Our results recover those of recent work of Chakraborti for k = 2 k = 2 .