Genus of the Cartesian product of triangles

Supplementary material for the paper Genus of the Cartesian product of triangles by M. Kotrbčík and T. Pisanski.

We denote the Cartesian product of n triangles by G_n

G₂:
We provide selected embeddings of G₂ related to Theorem 3.8, Table 5, and Table 6. In particular:

The list of all embeddings of G₂ with genus 1 and their corresponding face distributions; there are 330 such embeddings. These are the embeddings from the first case of Theorem 3.8.
The list of all embeddings of G₂ with genus 2 and their corresponding face distributions; there are 46 908 such embeddings. These are the embeddings from the second case of Theorem 3.8.
The list of all distinct extended face distributions of embeddings of G₂ with genus 1 and their corresponding rotation schemes; there are 7 such extended face distributions. These are the embeddings given in Table 6.
The list of all distinct extended face distributions of embeddings of G₂ with genus 1 and their corresponding rotation schemes of genus 2; there are 146 such extended face distributions.

G₃:
We provide all embeddings related to Theorem 3.9 and Table 7. In particular:

The list of 1319 pairwise distinct extended face distributions of embeddings of G₃ with genus 7 and their corresponding rotation schemes. These are the embeddings proving Theorem 3.9. Note that there may exist also other extended face distributions.
The list of selected 7 extended face distributions of embeddings of G₃ with genus 7 and their corresponding rotation schemes. These are the embeddings given in Table 7.

G₄:
We provide all embeddings related to Theorem 3.7, Table 3, Table 4. In particular:

The list of more than 10 000 embeddings of G₄ with genus 37 and their corresponding extended face distributions. These are the embeddings proving Theorem 3.7. Note that it is very likely that there are other extended face distributions of orientable embeddings of G₄ with genus 37.
The rotation schemes for the embedding from Table 4, whose extended face distribution is presented in Table 3. The embedding is given in two formats, one with vertices represented by integers, and the other, where vertices are represented by words over {0,1,2}. The representation by words is identical to that used in Table 4. For the description of representations see below.

Formats used.

Rotation schemes:
We use the following format for all rotation schemes with the single exception of the embedding from Table 4.

Each embedding is given by n+2 lines, where n is the number of vertices of the graph. The lines are as follows.

1st line: a single number n, the number of vertices of the graph;
2nd line: the string "genus: X number of faces: Y" where X is the genus of the embedding and Y is the number of faces of the embedding;

followed by n lines giving the rotations. Each of these lines has the form
i-th line: the degree of the i-th vertex followed by the rotation at the vertex. The rotation is given as a space-separated permutation of neighbours of the vertex.

For example, if the i-th line contains the string "4 0 1 3 2" it means that the vertex i has degree 4, the neighbours of the i-th vertex are 0, 1, 2, 3, and that the rotation at the i-th vertex is (0,1,3,2).

The vertices are numbered from zero.

The exceptional embedding of G₄, given in Table 4, uses the following format. The graph G₄ has 81 vertices and there are 81 lines, one for each vertex. The i-th line has the form

c_i: n₁, n₂, n₃, n₄, n₅, n₆, n₇, n₈

where c_i is the word corresponding to the coordinates of i-th vertex (see below) and n₁, ..., n₈ are the words corresponding to the neighbours of the i-th vertex and the rotation at i is (n₁, ..., n₈). (The words in the rotation are separated by a colon followed by a whitespace.)

We represent the coordinates of vertices as words over {0,1,2} as follows. The representation of i-th vertex is the reverse of fixed-length base-3 representation of the number i, where the length is n if the graph is G_n.

For example, in G₄ for the vertex number 7 we have 7 = 0021, so the vertex 7 is represented by the string 1200. For convenience, the translation tables for n = 2, 3, and 4 are provided.

Face distributions:

Each face distribution is given by two lines, each containing precisely 27 integers.

The i-th integer in the first line is the number of faces of length i+2. (All our graphs are simple and thus cannot contain faces of length 1 and 2. Therefore, the numbers of faces with length 1 and 2 are ommited).

The i-th integer in the second line is the number of repetitive faces among the faces of length i+2. A face is repetitive if it contains any vertex more than once.