This non-compact hyperbolic tesselation uses the euclidean tiling that in the sense of an infinite horohedron as cell.

Incidence matrix according to Dynkin symbol

x4o3x3x3*b   (N,M → ∞)

. . . .    | 24NM |    2    2    2 |   1    2    2   1   1   2 |  1  1   2  1
-----------+------+----------------+---------------------------+-------------
x . . .    |    2 | 24NM    *    * |   1    1    1   0   0   0 |  1  1   1  0
. . x .    |    2 |    * 24NM    * |   0    1    0   1   0   1 |  1  0   1  1
. . . x    |    2 |    *    * 24NM |   0    0    1   0   1   1 |  0  1   1  1
-----------+------+----------------+---------------------------+-------------
x4o . .    |    4 |    4    0    0 | 6NM    *    *   *   *   * |  1  1   0  0
x . x .    |    4 |    2    2    0 |   * 12NM    *   *   *   * |  1  0   1  0
x . . x    |    4 |    2    0    2 |   *    * 12NM   *   *   * |  0  1   1  0
. o3x .    |    3 |    0    3    0 |   *    *    * 8NM   *   * |  1  0   0  1
. o . x3*b |    3 |    0    0    3 |   *    *    *   * 8NM   * |  0  1   0  1
. . x3x    |    6 |    0    3    3 |   *    *    *   *   * 8NM |  0  0   1  1
-----------+------+----------------+---------------------------+-------------
x4o3x .    ♦   24 |   24   24    0 |   6   12    0   8   0   0 | NM  *   *  *
x4o . x3*b ♦   24 |   24    0   24 |   6    0   12   0   8   0 |  * NM   *  *
x . x3x    ♦   12 |    6    6    6 |   0    3    3   0   0   2 |  *  * 4NM  *
. o3x3x3*b ♦   3M |    0   3M   3M |   0    0    0   M   M   M |  *  *   * 8N

or
. . . .     | 12NM |    2    4 |   1    4   2   2 |  2   2  1
------------+------+-----------+------------------+----------
x . . .     |    2 | 12NM    * |   1    2   0   0 |  2   1  0
. . x .   & |    2 |    * 24NM |   0    1   1   1 |  1   1  1
------------+------+-----------+------------------+----------
x4o . .     |    4 |    4    0 | 3NM    *   *   * |  2   0  0
x . x .   & |    4 |    2    2 |   * 12NM   *   * |  1   1  0
. o3x .   & |    3 |    0    3 |   *    * 8NM   * |  1   0  1
. . x3x     |    6 |    0    6 |   *    *   * 4NM |  0   1  1
------------+------+-----------+------------------+----------
x4o3x .   & ♦   24 |   24   24 |   6   12   8   0 | NM   *  *
x . x3x     ♦   12 |    6   12 |   0    6   0   2 |  * 2NM  *
. o3x3x3*b  ♦   3M |    0   6M |   0    0  2M   M |  *   * 4N

6-coloring of x . . . edges of x4o3x3x3*b into gy/gb,vr/vb,or/oy
= 3-coloring of hexagons of x4o3x3x3*b into r,y,b

12NM    *    * |   1   1    2    2   0   0    0   0   0 |   1   1   1    2    2    2   0   0    0   0 |   2   1   1   0  1  ry
   * 12NM    * |   0   0    2    0   1   1    2   0   0 |   0   1   0    2    0    2   1   1    2   0 |   2   1   0   1  1  rb
   *    * 12NM |   0   0    0    2   0   0    2   1   1 |   0   0   1    0    2    2   0   1    2   1 |   2   0   1   1  1  yb
---------------+----------------------------------------+---------------------------------------------+-------------------
   2    0    0 | 6NM   *    *    *   *   *    *   *   * |   1   0   0    2    0    0   0   0    0   0 |   2   1   0   0  0  gy
   2    0    0 |   * 6NM    *    *   *   *    *   *   * |   1   0   0    0    2    0   0   0    0   0 |   2   0   1   0  0  vr
   1    1    0 |   *   * 24NM    *   *   *    *   *   * |   0   1   0    1    0    1   0   0    0   0 |   1   1   0   0  1  r
   1    0    1 |   *   *    * 24NM   *   *    *   *   * |   0   0   1    0    1    1   0   0    0   0 |   1   0   1   0  1  y
   0    2    0 |   *   *    *    * 6NM   *    *   *   * |   0   0   0    2    0    0   1   0    0   0 |   2   1   0   0  0  gb
   0    2    0 |   *   *    *    *   * 6NM    *   *   * |   0   0   0    0    0    0   1   0    2   0 |   2   0   0   1  0  or
   0    1    1 |   *   *    *    *   *   * 24NM   *   * |   0   0   0    0    0    1   0   1    1   0 |   1   0   0   1  1  b
   0    0    2 |   *   *    *    *   *   *    * 6NM   * |   0   0   0    0    2    0   0   0    0   1 |   2   0   1   0  0  vb
   0    0    2 |   *   *    *    *   *   *    *   * 6NM |   0   0   0    0    0    0   0   0    2   1 |   2   0   0   1  0  oy
---------------+----------------------------------------+---------------------------------------------+-------------------
   4    0    0 |   2   2    0    0   0   0    0   0   0 | 3NM   *   *    *    *    *   *   *    *   * |   2   0   0   0  0  gy-vr-gy-vr
   3    3    0 |   0   0    6    0   0   0    0   0   0 |   * 4NM   *    *    *    *   *   *    *   * |   0   1   0   0  1  r
   3    0    3 |   0   0    0    6   0   0    0   0   0 |   *   * 4NM    *    *    *   *   *    *   * |   0   0   1   0  1  y
   2    2    0 |   1   0    2    0   1   0    0   0   0 |   *   *   * 12NM    *    *   *   *    *   * |   1   1   0   0  0  gy-r-gb-r
   2    0    2 |   0   1    0    2   0   0    0   1   0 |   *   *   *    * 12NM    *   *   *    *   * |   1   0   1   0  0  vr-y-vb-y
   1    1    1 |   0   0    1    1   0   0    1   0   0 |   *   *   *    *    * 24NM   *   *    *   * |   1   0   0   0  1  ryb
   0    4    0 |   0   0    0    0   2   2    0   0   0 |   *   *   *    *    *    * 3NM   *    *   * |   2   0   0   0  0  gb-or-gb-or
   0    3    3 |   0   0    0    0   0   0    6   0   0 |   *   *   *    *    *    *   * 4NM    *   * |   0   0   0   1  1  b
   0    2    2 |   0   0    0    0   0   1    2   0   1 |   *   *   *    *    *    *   *   * 12NM   * |   1   0   0   1  0  or-b-oy-b
   0    0    4 |   0   0    0    0   0   0    0   2   2 |   *   *   *    *    *    *   *   *    * 3NM |   2   0   0   0  0  vb-oy-vb-oy
---------------+----------------------------------------+---------------------------------------------+-------------------
   8    8    8 |   4   4    8    8   4   4    8   4   4 |   2   0   0    4    4    8   2   0    4   2 | 3NM   *   *   *  *  sirco
   6    6    0 |   3   0   12    0   3   0    0   0   0 |   0   2   0    6    0    0   0   0    0   0 |   * 2NM   *   *  *  r-hip
   6    0    6 |   0   3    0   12   0   0    0   3   0 |   0   0   2    0    6    0   0   0    0   0 |   *   * 2NM   *  *  y-hip
   0    6    6 |   0   0    0    0   0   3   12   0   3 |   0   0   0    0    0    0   0   2    6   0 |   *   *   * 2NM  *  b-hip
  3M   3M   3M |   0   0   6M   6M   0   0   6M   0   0 |   0   M   M    0    0   6M   0   M    0   0 |   *   *   *   * 4N  that