Levi Segal

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math.CO2026

Improved Explicit and Algorithmic Lower Bounds for r(5,t)

Amrit Kandasamy, Lawrence Zhou, Levi Segal

We modify a construction of Bradač \cite{Bradac26} using a vertex ordering to give, for every prime power $q$ , an explicit $K_{5}$ free graph with more than $q^{7}$ vertices and indepe…

math.CO2026

Polynomial Length Cycles In Tough Graphs

Levi Segal, Jacques Verstraete

We prove that every $n$ -vertex $15$ -tough graph contains a cycle of length $Ω (n^{1/20})$ .

math.CO2026

Logarithmic Circumference In Tough Graphs

Levi Segal, Jacques Verstraete

For every real $t > 0$ , we prove that every $2$ -connected $t$ -tough graph contains a cycle of length at least $ℓ$ whenever $n \leq ℓ (1 + 1/ t)^{⌊ ℓ /2 ⌋ - 1}$ . This…

math.CO2026

Contraction Containment in Labeled Trees: Support Counts, Collision Cores, and Star Avoidance

Levi Segal

We study whether a labeled tree can be recovered from the numbers of larger labeled trees that contract to it. We first characterize containment using the edge splits induced on th…