Sparse spanning $k$-connected subgraphs in tournaments

TL;DR

This paper proves that in strongly k-connected tournaments, there exists a spanning subgraph with at most kn + O(k^2 log k) arcs that maintains strong k-connectivity, answering a longstanding question.

Contribution

It establishes an explicit upper bound on the size of sparse strongly k-connected spanning subgraphs in tournaments, resolving a question posed in 2009.

Findings

Existence of sparse strongly k-connected spanning subgraphs with at most kn + 750k^2 log(k+1) arcs.

Provides an explicit function g(k) for the size bound.

Answers a 2009 open problem in tournament connectivity.

Abstract

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