Konig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell . As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed.
Bibliographical noteFunding Information:
This project is supported by the 3 M non-tenured faculty program and the National Science Foundation #1362256. We would like to thank Dr Zhiqiang Fang for folding resistance measurement, Dr Yuhuang Wang and Mr Brendan Meany for UV-Vis spectrum measurements, and Dr Jeremy Munday and Mr Joe Murphy for angular light scattering measurements. The Maryland Nanocenter and its NispLab are also greatly acknowledged. Dr Jianhua Zou and Dr Junbiao Peng would like to thank 973 chief project (No. 2015CB655000), the National Natural Science Foundation of China (No. 61574061 and 61574062), Science and Technology Project of Guangdong Province (No. 2014B090915004, 2014B090916002 and 2015B090914003). Dr Tao acknowledges the funding of the State Key Laboratory of Pulp and Paper Engineering, South China University of Technology (Grant No. 201233 and 2014C17), and the funding of the Guangdong Province Industrial Science and technology projects (Grant No. 2013B010406003).
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics