Genus of the cartesian product of triangles

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Authors

KOTRBČÍK Michal PISANSKI Tomaz

Year of publication 2015
Type Article in Periodical
Magazine / Source Electronic Journal of Combinatorics
MU Faculty or unit

Faculty of Informatics

Citation
Field General mathematics
Keywords graph; cartesian product; genus; embedding; triangle; symmetric embedding; Cayley graph; Cayley map; genus range; group
Description We investigate the orientable genus of G(n), the cartesian product of n triangles, with a particular attention paid to the two smallest unsolved cases n = 4 and 5. Using a lifting method we present a general construction of a low -genus embedding of G(n) using a low-genus embedding of G(n-1). Combining this method with a computer search and a careful analysis of face structure we show that 30 <= gamma (G(4)) <= 37 and 133 <= gamma(G(5)) <= 190. Moreover, our computer search resulted in more than 1300 non isomorphic minimum -genus embeddings of G(3). We also introduce genus range of a group and (strong) symmetric genus range of a Cayley graph and of a group. The (strong) symmetric genus range of irredundant Cayley graphs of Z(p)(n) is calculated for all odd primes p.
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