Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number
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The biclique cover number (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 bc(G) ≥⌈log_2(mc(G^c)) ⌉≥⌈log_2(χ(G)) ⌉ for an arbitrary graph G, where χ(G) is the chromatic number of G and 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 ⌈log_2(mc(G^c)) ⌉ could be a strictly tighter lower bound of the biclique cover number than other existing lower bounds. We can also provide a bound of bc(G) with respect to the biclique partition number (bp) of G: bc(G) ≥⌈log_2(bp(G) + 1) ⌉ or bp(G) ≤ 2^bc(G) - 1 if G is co-chordal. Furthermore, we show that bc(G) ≤χ_r'(T_K^c), where G is a co-chordal graph such that each vertex is in at most two maximal independent sets and χ_r'(T_K^c) is the optimal edge-ranking number of a clique tree of G^c.
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