Non-repeated cycle lengths and Sidon sequences

TL;DR

This paper proves a conjecture about the maximum edges in 2-connected graphs with unique cycle lengths, linking it to the problem of finding maximum Sidon sequences, thus bridging graph theory and number theory.

Contribution

It establishes a proof for a conjecture on cycle lengths in graphs and connects this problem to the classical problem of Sidon sequences in number theory.

Findings

Proved the conjecture on maximum edges in 2-connected graphs without repeated cycle lengths.

Reduced the problem to the maximum Sidon sequence problem.

Provided a matched lower bound construction.

Abstract

We prove a conjecture of Boros, Caro, F\"uredi and Yuster on the maximum number of edges in a 2-connected graph without repeated cycle lengths, which is a restricted version of a longstanding problem of Erd\H{o}s. Our proof together with the matched lower bound construction of Boros, Caro, F\"uredi and Yuster show that this problem can be conceptually reduced to the seminal problem of finding the maximum Sidon sequences in number theory.