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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 topologicalconnectivity of its neighbourhood complex. We also work out conditions underwhich computing the topological connectivity of hom complex of a pair of graphscan be simplified. In particular, hom complex as a covariant functor acting ona double mapping cylinder of graphs is a homotopy pushout of hom complexfunctor applied to its subgraphs. We give applications of this result where thecomputation of hom complexes is simplified. Finally, we explain why doublemapping cylinder of graphs does not give a satisfactory definition of homotopypushout in the category of graphs.

Lov\'{a}sz' original lower bound: Getting tighter bounds and Reducing computational complexity