Sparse Spanning $k$-Connected Subgraphs in Tournaments

Abstract

In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that\nevery strongly $k$-connected $n$-vertex tournament contains a strongly\n$k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper,\nwe answer the question by showing that every strongly $k$-connected $n$-vertex\ntournament contains a strongly $k$-connected spanning subgraph with at most $kn\n+ 750k^2\\log(k+1)$ arcs.\n