Fedor Fomin gave an online talk on fixed-parameter algorithms to decide whether a 2-connected graph has a cycle whose length is at least k plus the bound given by the Dirac’s theorem or the Erdős-Gallai theorem at the Virtual Discrete Math Colloquium

On March 10, 2022, Fedor Fomin from the University of Bergen gave an online talk at the Virtual Discrete Math Colloquium on the parameterized complexity of determining whether a 2-connected graph has a cycle of length at least k plus the bound given by the Dirac’s theorem or the Erdős-Gallai theorem. The title of his talk was “Long cycles in graphs: Extremal Combinatorics meets Parameterized Algorithms“.

Fedor Fomin, Long cycles in graphs: Extremal Combinatorics meets Parameterized Algorithms

We examine algorithmic extensions of two classic results of extremal combinatorics. First, the theorem of Dirac from 1952 asserts that a 2-connected graph G with the minimum vertex degree d>1, is either Hamiltonian or contains a cycle of length at least 2d. Second, the theorem of Erdős-Gallai from 1959, states that a 2-connected graph G with the average vertex degree D>1, contains a cycle of length at least D.

We discuss the recent progress in parameterized complexity of computing long cycles “above” the guarantees established by these classical theorems: cycles of lengths at least 2d+k and D+k.

The talk is based on the joint works with Petr Golovach, Danil Sagunov, and Kirill Simonov.

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