  In this article, we give conditions on a graph under which the Lov\'{a}sz'
original bound of the graph can be improved by increasing the topological
connectivity of its neighbourhood complex. We also work out conditions under
which computing the topological connectivity of hom complex of a pair of graphs
can be simplified. In particular, hom complex as a covariant functor acting on
a double mapping cylinder of graphs is a homotopy pushout of hom complex
functor applied to its subgraphs. We give applications of this result where the
computation of hom complexes is simplified. Finally, we explain why double
mapping cylinder of graphs does not give a satisfactory definition of homotopy
pushout in the category of graphs.
