2026/08/03 by Omid Khormali
Mathematics · #math.CO #msc:05C35
arxiv created 2026/08/03 · arxiv updated 2026/08/04
The bipartite Turán number of a graph H, denoted ex(m, n; H), is the maximum number of edges in any H-free bipartite graph G = (A, B; E) with parts of size |A| = m and |B| = n. We study this problem for two families. For a tree T = T(r, s) with parts R and S of sizes |R| = r ≤ s = |S|, we prove (r - 1) n ≤ ex(m, n; T(r, s)) ≤ (r - 1) n + O(m) for n sufficiently large compared to m, r, and s, determining the leading-order term exactly (with the star case r=1 solved with an exact formula). For a star forest F = \bigcupi=1k Sdi with d1 ≥ ⋯ ≥ dk, we determine the exact value ex(m, n; F) = (k - 1) n + (dk - 1)(m - k + 1) for n sufficiently large, and characterize the unique extremal graph.