Tag: Discrete Mathematics and Combinatorics
Entropy bounds limit perfect matchings in bipartite hypergraphs
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in MathematicsWhat the study found The authors prove an upper bound on the number of A-perfect matchings in uniform bipartite hypergraphs with small maximum codegree. They also derive bounds for related counting problems in Latin squares and regular hypergraphs. Why the authors say this matters The study suggests that these bounds help quantify how many perfect…
Exact maximum diameter found for 2-dimensional simplicial complexes
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in MathematicsWhat the study found The study determines the exact maximum diameter for 2-dimensional abstract simplicial complexes on n vertices for every n. It also identifies an infinite sequence of explicit constructions that achieve this bound. Why the authors say this matters The authors present their result as an answer to a problem posed by Santos…
Combinatorial proofs established for remaining partition results
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in MathematicsWhat the study found The authors provide combinatorial proofs for the remaining results concerning two-colored partitions and overpartitions with constraints. Why the authors say this matters The abstract says this work addresses questions raised by Andrews and El Bachraoui about whether combinatorial proofs exist for these results, and it extends earlier partial work by the…
Tensor product formulas extend to two graph polynomials
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What the study found The authors define a tensor product for graphs embedded in pseudo-surfaces and use it to generalize and unify several existing tensor product formulas. They provide Brylawski-style formulas for both the Bollobás-Riordan polynomial and the Krushkal polynomial. Why the authors say this matters The study suggests that this framework brings together previously…
Extremal signed complete graphs with K2,2-minor-free negative subgraphs
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in MathematicsWhat the study found The authors characterize the extremal signed complete graphs that achieve the maximum and second maximum index when the negative-edge-induced subgraph is a K2,2-minor-free spanning subgraph of Kn. Why the authors say this matters The abstract says this work addresses an extremum problem for the index of a signed complete graph based…
Edge version of graph inducibility is determined by fractional independence number
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in MathematicsWhat the study found The study shows that the edge version of inducibility for any graph H satisfies ρ(H,m) = Θ(m^α_f(H)), where α_f(H) is the fractional independence number of H. The authors also give additional bounds and conjectures for paths and cycles. Why the authors say this matters The authors indicate that this result shifts…
