Abstract
The bicliqueB of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graphKB(G) of a graph G is the graph with the bicliques of G as its vertices where two vertices of KB(G) are adjacent if and only if they have a common vertex in G. In this paper, we disprove a conjecture of Groshaus and Montero concerning Helly property of biclique graphs (Groshaus and Montero, 2021) and two conjectures of Montero concerning vertex removal in biclique graphs (Montero, 2022). In addition, we confirm a conjecture of Montero about the structural characterization of biclique graphs in the same paper.
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