Complex of Complete Bipartite Subgraphs

we deflne a weU-known concept in the generality that we need here. 

Let G be a finite graph. Recall that is a simplex whose set of 

Note that if A, B) is a complete bipartite subgraph of G, and A C A, B G B, A, B then (A,B) is also a complete bipartite subgraph of G. This verifies that Bip (G) is actually a subcomplex. 

Example 9.19. Recall that Kn denotes the complete graph on n vertices, and Ln denotes the string graph on n vertices, i.e., we have V L ) = [n], and 

The prodsimplicial complex Bip (G) is our first example of the so-called Hom(-, -)-construction, namely, as a polyhedral complex it is isomorphic to Horn K2, G). 

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