Typical properties of countable graphs: flows and bridges

Abstract

This paper has two aims. First, we study nowhere-zero flows in countably infinite graphs, observing the different roles played by two types of bridge. Second, we ask how typical graphs with such bridges are. Since there is no canonical probability measure on countable graphs, we explore an analytic approach and study which graph classes are nowhere dense and which are meager for two different definitions of a distance. We characterize in this sense graphs with a bridge (separating a finite component, or two infinite ones), $D$-edge-colorable graphs, etc.