Looks like 2 coincident rhombitrihexagonal tilings (rothat), and indeed vertices, edges, {4} and {6} all coincide by pairs.

Incidence matrix according to Dynkin symbol

β6β3x   (N → ∞)

demi( . . . (a)) | 6N  * |  1  2  0  1 | 1 0  1  2
demi( . . . (b)) |  * 6N |  1  0  2  1 | 0 1  1  2
-----------------+-------+-------------+----------
both( . . x    ) |  1  1 | 6N  *  *  * | 0 0  1  1
sefa( s6s . (a)) |  2  0 |  * 6N  *  * | 1 0  0  1
sefa( s6s . (b)) |  0  2 |  *  * 6N  * | 0 1  0  1
sefa( . β3x    ) |  1  1 |  *  *  * 6N | 0 0  1  1
-----------------+-------+-------------+----------
      s6s . (a)  ♦  6  0 |  0  6  0  0 | N *  *  *
      s6s . (b)  ♦  0  6 |  0  0  6  0 | * N  *  *
      . β3x      ♦  3  3 |  3  0  0  3 | * * 2N  *
sefa( β6β3x    ) |  2  2 |  1  1  1  1 | * *  * 6N

or
both( . . . )    | 6N |  1  2  1 | 1 1  2
-----------------+----+----------+-------
both( . . x )    |  2 | 3N  *  * | 0 1  1
sefa( s6s . )  & |  2 |  * 6N  * | 1 0  1
sefa( . β3x )    |  2 |  *  * 3N | 0 1  1
-----------------+----+----------+-------
      s6s .    & ♦  6 |  0  6  0 | N *  *
      . β3x      ♦  6 |  3  0  3 | * N  *
sefa( β6β3x )    |  4 |  1  2  1 | * * 3N

starting figure: x6x3x

x3β6x   (N → ∞)

demi( . . . (a)) | 6N  * |  1  1  0  1  1  0 |  1 1 0  1  1
demi( . . . (b)) |  * 6N |  1  0  1  1  0  1 |  1 0 1  1  1
-----------------+-------+-------------------+-------------
both( x . .    ) |  1  1 | 6N  *  *  *  *  * |  1 0 0  1  0
demi( . . x (a)) |  2  0 |  * 3N  *  *  *  * |  0 1 0  1  0
demi( . . x (b)) |  0  2 |  *  * 3N  *  *  * |  0 0 1  1  0
sefa( x3β .    ) |  1  1 |  *  *  * 6N  *  * |  1 0 0  0  1
sefa( . s6x (a)) |  2  0 |  *  *  *  * 3N  * |  0 1 0  0  1
sefa( . s6x (b)) |  0  2 |  *  *  *  *  * 3N |  0 0 1  0  1
-----------------+-------+-------------------+-------------
      x3β .      ♦  3  3 |  3  0  0  3  0  0 | 2N * *  *  *
      . s6x (a)  ♦  6  0 |  0  3  0  0  3  0 |  * N *  *  *
      . s6x (b)  ♦  0  6 |  0  0  3  0  0  3 |  * * N  *  *
both( x . x    ) |  2  2 |  2  1  1  0  0  0 |  * * * 3N  *
sefa( x3β6x    ) |  2  2 |  0  0  0  2  1  1 |  * * *  * 3N

or
both( . . . )    | 12N |  1  1  1  1 |  1  1  1  1
-----------------+-----+-------------+------------
both( x . . )    |   2 | 6N  *  *  * |  1  0  1  0
both( . . x )    |   2 |  * 6N  *  * |  0  1  1  0
sefa( x3β . )    |   2 |  *  * 6N  * |  1  0  0  1
sefa( . s6x )  & |   2 |  *  *  * 6N |  0  1  0  1
-----------------+-----+-------------+------------
      x3β .      ♦   6 |  3  0  3  0 | 2N  *  *  *
      . s6x    & ♦   6 |  0  3  0  3 |  * 2N  *  *
both( x . x )    |   4 |  2  2  0  0 |  *  * 3N  *
sefa( x3β6x )    |   4 |  0  0  2  2 |  *  *  * 3N

starting figure: x3x6x