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Discrete Mathematics and Combinatorics

Mathematics of discrete structures, including graph theory, combinatorial optimization, coding theory, discrete geometry, and applications to computer science and operations research

4 papers in this specialization

Paperzilla Papers in Discrete Mathematics and Combinatorics

AN ALMOST CONSTANT LOWER BOUND OF THE ISOPERIMETRIC COEFFICIENT IN THE KLS CONJECTURE

This paper proves an almost constant lower bound for the isoperimetric coefficient in the KLS conjecture, improving upon previous bounds with better dimension dependency. This has implications for other related conjectures like Bourgain's slicing conjecture and the thin-shell conjecture, plus potential impacts on concentration inequalities and mixing time bounds for log-concave measures.

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s00039-021-00558-4.pdf

Jul 14, 11:21 AM

Hopf monoids and generalized permutahedra

The paper introduces a Hopf algebraic structure on generalized permutahedra, proving they are the universal family of polytopes with this structure. It presents a cancellation-free antipode formula and demonstrates how this framework unifies classical results in combinatorics related to graphs, matroids, posets, and inversion of power series, offering potential for new discoveries.

Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups

The paper classifies factorizations of almost simple groups where at least one factor is solvable. It then applies this classification to characterize s-arc-transitive Cayley graphs of solvable groups, discovering that, except for cycles, all non-bipartite connected 3-arc-transitive Cayley graphs are covers of the Petersen or Hoffman-Singleton graphs.

Counting quadrant walks via Tutte's invariant method

This paper adapts Tutte's invariant method to analyze quadrant walks, proving algebraicity for several models, including Gessel's. It introduces a weaker invariant notion, demonstrating D-algebraicity for nine non-D-finite models with decoupling functions, using a novel integral-free expression for their generating function.