Looks like 2 coincident triangular-hexagonal tilings (that), and indeed vertices, edges, and {6} all coincide by pairs.

Incidence matrix according to Dynkin symbol

β3x6o   (N → ∞)

both( . . . ) | 6N |  2  2 |  2 1 1
--------------+----+-------+-------
both( . x . ) |  2 | 6N  * |  1 1 0
sefa( β3x . ) |  2 |  * 6N |  1 0 1
--------------+----+-------+-------
      β3x .   ♦  6 |  3  3 | 2N * *
both( . x6o ) |  6 |  6  0 |  * N *
sefa( β3x6o ) |  6 |  0  6 |  * * N

starting figure: x3x6o