Warmth and edge spaces of graphs

Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph G is always less than three plus the connectivity of Hom(K2,G). We succeed in establishing a first nontrivial case of the conjecture, by showing that ζ(G)≤3 if Hom(K2,G) has an infinite first homology group. We also calculate warmth for a family of 'twisted toroidal' graphs that are important extremal examples in the context of Hom complexes. Finally we show that ζ(G)≤n−1 if a graph G does not have the complete bipartite graph Ka,b for a+b=n. This provides an analogue for a similar result in the context of Hom complexes.