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 $30\le \gamma(G_4) \le 37$ and $133 \le\gamma(G_5) \le 190$. 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 $p$.
@article{10_37236_2951,
author = {Michal Kotrb\v{c}{\'\i}k and Toma\v{z} Pisanski},
title = {Genus of the {Cartesian} product of triangles},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {4},
doi = {10.37236/2951},
zbl = {1323.05108},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2951/}
}
TY - JOUR
AU - Michal Kotrbčík
AU - Tomaž Pisanski
TI - Genus of the Cartesian product of triangles
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/2951/
DO - 10.37236/2951
ID - 10_37236_2951
ER -
%0 Journal Article
%A Michal Kotrbčík
%A Tomaž Pisanski
%T Genus of the Cartesian product of triangles
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/2951/
%R 10.37236/2951
%F 10_37236_2951
Michal Kotrbčík; Tomaž Pisanski. Genus of the Cartesian product of triangles. The electronic journal of combinatorics, Tome 22 (2015) no. 4. doi: 10.37236/2951
