A characterization of hypergraphs that achieve equality in the Chvátal–McDiarmid Theorem

For k ≥ 2 , let H be a k -uniform hypergraph on n vertices and m edges. The transversal number τ ( H ) of H is the minimum number of vertices that intersect every edge. Chvátal and McDiarmid (1992) proved that τ ( H ) ≤ ( n + ⌊ k 2 ⌋ m ) / ( ⌊ 3 k 2 ⌋ ) . When k = 3 , the connected hypergraphs that achieve equality in the Chvátal–McDiarmid Theorem were characterized by Henning and Yeo (2008). In this paper, we characterize the connected hypergraphs that achieve equality in the Chvátal–McDiarmid Theorem for k = 2 and for all k ≥ 4 .