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New computational results on a conjecture of Jacobsthal
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abstract
Jacobsthal's conjecture has been disproved by counterexample a few years ago. We continue to verify this conjecture on a larger scale. For this purpose, we implemented an extension of the Greedy Permutation Algorithm and computed the maximum Jacobsthal function for the product of $k$ primes up to $k=43$. We have found various new counterexamples. Their pattern seems to imply that the conjecture of Jacobsthal only applies to several small $k$. Our results raise further questions for discussion. In addition to this paper, we provide exhaustive information about all covered sequences of the appropriate maximum lengths in ancillary files.
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Domination in direct products of complete graphs
For unitary Cayley graphs X_n, the author constructs n with arbitrarily many prime factors satisfying γ_t(X_n) ≤ g(n) - 16, and classifies all products of complete graphs with domination number t+2.
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