Given a simple bipartite graph G and an integer t≥2, we derive a formula for the maximum number of edges in a subgraph H of G so that H contains no node of degree larger than t and H contains no complete bipartite graph Kt,t as a subgraph. In the special case t=2 this fomula was proved earlier by Király (Square-free 2-matching in bipartite graphs, Technical Report of Egerváry Research Group, TR-2001013, November 1999 (www.cs.elte.hu/egres)), sharpening a result of Hartvigsen (in: G. Cornuejols, R. Burkard, G.J. Woeginger (Eds.), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 1610, Springer, Berlin, 1999, pp. 234-240). For any integer t≥2, we also determine the maximum number of edges in a subgraph of G that contains no complete bipartite graph, as a subgraph, with more than t nodes. The proofs are based on a general min-max result of Frank and Jordán (J. Combin. Theory Ser. B 65(1) (1995) 73) concerning crossing bi-supermodular functions.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics