Tension continuous maps—Their structure and applications

We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly T T  mappings). The existence of a T T  mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see Hell and Nešetřil (2004) [10]). In this paper we study the relationship of the homomorphism and T T orders. We stress the similarities and the differences in both deterministic and random settings. Particularly, we prove that T T order is universal and investigate graphs for which homomorphisms and T T mappings coincide (so-called homotens graphs). In the course of our study, we prove a new Ramsey-type theorem, which may be of independent interest. We solve a problem asked in [Matt DeVos, Jaroslav Nešetřil, André Raspaud, On edge-maps whose inverse preserves flows and tensions, in: J.A. Bondy, J. Fonlupt, J.-L. Fouquet, J.-C. Fournier, J.L. Ramirez Alfonsin (Eds.), Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge, in: Trends in Mathematics, Birkhäuser, 2006].