Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Bipartite Coverings and the Chromatic Number. | Bipartite Coverings and the Chromatic Number Dhruv Mubayi Department of Mathematics Statistics and Computer Science University of Illinois Chicago IL 60607 USA mubayi@ Sundar Vishwanathan Department of Computer Science Indian Institute of Technology Mumbai India 400076 sundar@ Submitted Feb 14 2009 Accepted Nov 17 2009 Published Nov 30 2009 Abstract Consider a graph G with chromatic number k and a collection of complete bipartite graphs or bicliques that cover the edges of G. We prove the following two results If the bipartite graphs form a partition of the edges of G then their number is at least 2log2 k. This is the first improvement of the easy lower bound of log2 k while the Alon-Saks-Seymour conjecture states that this can be improved to k 1. The sum of the orders of the bipartite graphs in the cover is at least 1 o 1 k log2 k. This generalizes in asymptotic form a result of Katona and Sze-meredi who proved that the minimum is k log2 k when G is a clique. 1 Introduction It is a well-known fact that the minimum number of bipartite graphs needed to cover the edges of a graph G is flog x G l where x G is the chromatic number of G all logs are to the base 2 . Two classical theorems study related questions. One is the Graham-Pollak theorem 1 which states that the minimum number of complete bipartite graphs needed to partition E Kk is k 1. Another is the Katona-Szemeredi theorem 4 which states that the minimum of the sum of the orders of a collection of complete bipartite graphs that cover E Kk is k log k. Both of these results are best possible. An obvious way to generalize these theorems is to ask whether the same results hold for any G with chromatic number k . Conjecture 1 Alon - Saks - Seymour The minimum number of complete bipartite graphs needed to partition the edge set of a graph G with chromatic number k is k 1. Note that every graph has a partition of this size simply by taking a proper coloring V1 . Vk and letting the ith .
