Bipartite coverings and the chromatic number
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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: $\bullet$ If the bipartite graphs form a partition of the edges of $G$, then their number is at least $2^{\sqrt{\log_2 k}}$. This is the first improvement of the easy lower bound of $\log_2 k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$. $\bullet$ The sum of the orders of the bipartite graphs in the cover is at least $(1-o(1))k\log_2 k$. This generalizes, in asymptotic form, a result of Katona and Szemerédi who proved that the minimum is $k\log_2 k$ when $G$ is a clique.
@article{10_37236_272,
author = {Dhruv Mubayi and Sundar Vishwanathan},
title = {Bipartite coverings and the chromatic number},
journal = {The electronic journal of combinatorics},
year = {2009},
volume = {16},
number = {1},
doi = {10.37236/272},
zbl = {1186.05099},
url = {http://geodesic.mathdoc.fr/articles/10.37236/272/}
}
Dhruv Mubayi; Sundar Vishwanathan. Bipartite coverings and the chromatic number. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/272
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