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Loop space construction of bigraphs and box complexes

Dochtermann introduced the loop space construction of a based graph $(G,v)$ whose basepoint is a looped vertex. He showed that the complex $C(Ω(G,v))$ is homotopy equivalent to the loop space $Ω(C(G),v)$ of $C(G)$. Here we write $C(G)$ to mean the clique complex of the maximal reflexive subgraph of $G$. In this paper, we consider its bigraph version. A bigraph is a graph equipped with its 2-coloring. We introduce the loop space construction $Ω_{/K_2}(X,x)$ of a based bigraph $(X,x)$. This is a graph such that $C(Ω_{/K_2}(X,x))$ is homotopy equivalent to the loop space of the box complex $B_{/K_2}(X)$ of the bigraph. As a result, we have alternative proofs of some results of Matsushita and Schultz.