Bipartite Turán Problem on Complete Bipartite Graphs

Abstract

The bipartite Turán number of a bipartite graph H, denoted by ex(m, n; H), is the maximum number of edges in subgraphs of K_m,n containing no copies of H. In this paper, we determine the bipartite Turán number of the complete bipartite graphs ex(m; n; K_s,t) = mn − (⌊(n − t)/s⌋(m − s) + m + n − s − t + 1) when n ≥ s + t and t ≥ s, if one of the following is satisfied: n = s + t; m ≤ s + (t − 1)/(⌊(n − t)/s⌋ + 1); s∣(n − t) and m ≤ sn/(n − t); being m ≥ 2s if m = sn/(n − t).