Julius Tranquilli | Searcharxiv

arXiv2026

A 60-Vertex Lower Bound for Cubic Bipartite Counterexamples to the Erd\H{o}s-Gy\'arf\'as Conjecture

A certified exhaustive computation shows that every simple cubic bipartite graph on at most 58 vertices contains a cycle of length 4, 8, or 16. Consequently, any cubic bipartite counterexample to the Erdos-Gyarfas conjecture has at least 60 vertices, improving the established published lower bound for this class from 30 to 60. The proof begins with a Moore-bound observation: below 62 vertices, a cubic bipartite graph avoiding 4- and 8-cycles must contain a 6-cycle. Viewing the graph as the Levi graph of a linear symmetric v3-configuration turns this 6-cycle into a Berge triangle. Up to symmetry, only two rooted extensions are possible. A complete restricted-growth search on at most 29 points exhausts both search trees. The computation is checked by two separately implemented exact procedures using different C16 oracles and by a static witness certificate. Source code, certificates, and reproduction instructions are archived with the paper.

math.CO↗

arXiv2025

Formal State-Machine Models for Uniswap v3 Concentrated-Liquidity AMMs: Priced Timed Automata, Finite-State Transducers, and Provable Rounding Bounds

Concentrated-liquidity automated market makers (CLAMMs), as exemplified by Uniswap v3, are now a common primitive in decentralized finance frameworks. Their design combines continuous trading on constant-function curves with discrete tick boundaries at which liquidity positions change and rounding effects accumulate. While there is a body of economic and game-theoretic analysis of CLAMMs, there is negligible work that treats Uniswap v3 at the level of formal state machines amenable to model checking or theorem proving. In this paper we propose a formal modeling approach for Uniswap v3-style CLAMMs using (i) networks of priced timed automata (PTA), and (ii) finite-state transducers (FST) over discrete ticks. Positions are treated as stateful objects that transition only when the pool price crosses the ticks that bound their active range. We show how to encode the piecewise constant-product invariant, fee-growth variables, and tick-crossing rules in a PTA suitable for tools such as UPPAAL, and how to derive a tick-level FST abstraction for specification in TLA+. We define an explicit tick-wise invariant for a discretized, single-tick CLAMM model and prove that it is preserved up to a tight additive rounding bound under fee-free swaps. This provides a formal justification for the "$\epsilon$-slack" used in invariance properties and shows how rounding enters as a controlled perturbation. We then instantiate these models in TLA+ and use TLC to exhaustively check the resulting invariants on structurally faithful instances, including a three-tick concentrated-liquidity configuration and a bounded no-rounding-only-arbitrage property in a bidirectional single-tick model. We discuss how these constructions lift to the tick-wise structure of Uniswap v3 via virtual reserves, and how the resulting properties can be phrased as PTA/TLA+ invariants about cross-tick behaviour and rounding safety.

cs.LO↗