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# One n Remains to Settle the Tree Conjecture

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LIPIcs.STACS.2024.28.pdf
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## Cite As

Jack Dippel and Adrian Vetta. One n Remains to Settle the Tree Conjecture. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.28

## Abstract

In the famous network creation game of Fabrikant et al. [Fabrikant et al., 2003] a set of agents play a game to build a connected graph. The n agents form the vertex set V of the graph and each vertex v ∈ V buys a set E_v of edges inducing a graph G = (V,⋃_{v∈V} E_v). The private objective of each vertex is to minimize the sum of its building cost (the cost of the edges it buys) plus its connection cost (the total distance from itself to every other vertex). Given a cost of α for each individual edge, a long-standing conjecture, called the tree conjecture, states that if α > n then every Nash equilibrium graph in the game is a spanning tree. After a plethora of work, it is known that the conjecture holds for any α > 3n-3. In this paper we prove the tree conjecture holds for α > 2n. This reduces by half the open range for α with only (n-3, 2n) remaining in order to settle the conjecture.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Algorithmic game theory
• Theory of computation → Social networks
##### Keywords
• Algorithmic Game Theory
• Network Creation Games
• Tree Conjecture

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## References

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