Homotopy groups of Hom complexes of graphs

The notion of ×-homotopy from [Anton Dochtermann, Hom complexes and homotopy theory in the category of graphs, European J. Combin., in press] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Hom ∗ ( G , H ) with the homotopy groups of Hom ∗ ( G , H I ) . Here Hom ∗ ( G , H ) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and H I is the graph of based paths in H. As a corollary it is shown that π i ( Hom ∗ ( G , H ) ) ≅ [ G , Ω i H ] × , where ΩH is the graph of based closed paths in H and [ G , K ] × is the set of ×-homotopy classes of pointed graph maps from G to K. This is similar in spirit to the results of [Eric Babson, Hélène Barcelo, Mark de Longueville, Reinhard Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin. 24 (1) (2006) 31–44], where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.