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- W4322716025 abstract "The biclique cover number $(text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $text{bc}(G) geq lceil log_2(text{mc}(G^c)) rceil geq lceil log_2(chi(G)) rceil$ for an arbitrary graph $G$, where $chi(G)$ is the chromatic number of $G$ and $text{mc}(G^c)$ is the number of maximal cliques of the complementary graph $G^c$, i.e., the number of maximal independent sets of $G$. We also show that $lceil log_2(text{mc}(G^c)) rceil$ could be a strictly tighter lower bound of the biclique cover number than other existing lower bounds. We can also provide a bound of $text{bc}(G)$ with respect to the biclique partition number ($text{bp}$) of $G$: $text{bc}(G) geq lceil log_2(text{bp}(G) + 1) rceil$ or $text{bp}(G) leq 2^{text{bc}(G)} - 1$ if $G$ is co-chordal. Furthermore, we show that $text{bc}(G) leq chi_r'(T_{{K}^c})$, where $G$ is a co-chordal graph such that each vertex is in at most two maximal independent sets and $chi_r'({T}_{{K}^c})$ is the optimal edge-ranking number of a clique tree of $G^c$." @default.
- W4322716025 created "2023-03-03" @default.
- W4322716025 creator A5018115183 @default.
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- W4322716025 date "2023-02-24" @default.
- W4322716025 modified "2023-09-23" @default.
- W4322716025 title "Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number" @default.
- W4322716025 doi "https://doi.org/10.48550/arxiv.2302.12775" @default.
- W4322716025 hasPublicationYear "2023" @default.
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