Immersions of Directed Graphs in Tournaments

ABSTRACT Recently, Draganić, Munhá Correia, Sudakov and Yuster (2022) showed that every tournament on vertices contains a 1‐subdivision of a transitive tournament on vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let be the smallest integer such that any tournament on at least vertices must contain a 1‐immersion of a transitive tournament on vertices. We show that , which is clearly tight up to a multiplicative factor. If one insists on finding an immersion of a complete directed graph on vertices, then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out‐degree at least must contain a 2‐immersion of a complete digraph on vertices. This is again tight up to the value of and tight on the order of the paths in the immersion.