Definition
In a tiling two tiles A and B are neighbors of each other, if A∩B ≠ {} and A∩B ≠ A.

Conjecture

Comments
Otherwise there has to exist a tile in the tiling that is completely surrounded by two other tiles.
A tile with two properties is necessary:
(1) The tile admits a tiling of the plane.
(2) There exists a patch with 3 copies of the tile, where one copy is completely surrounded by the others.

As far as we know, only the V-Dragons meet these conditions.
Definition
A dragon patch consists of three V-Dragon tiles with an inner tile A, a tile B that is a 180°-rotation of tile A, and tile C that is a rotation of A by an angle ε < 30°.

For a proof of the conjecture it would be sufficientto show:
(a) Each patch of 3 tiles where one tile is completely surrounded by the others is a dragon patch.
(b) There exists no monohedral V-Dragon tiling in which a dragon patch occurs.

Note: It is easy to find a tile that can be completely surrounded by two copies of itself as long as we allow holes in this composition.

A monohedral patch formed by 4 tiles, where one tile is completely surrounded by the other 3 tiles is also called Tetrad.
Tetrads are known since the 1970s.
Up to simple transformations we only know one tetrad that can tile the plane. This tetrad consists of 4 congruent tetrahexes.
But it is not necessary that the whole patch can tile the plane, it is sufficient that the tile can tile the plane and that in this tiling occurs at least one tetrad.


The yellow tiles have only 3 neighbors.
The other tiles have 7 or 8 neighbors. The average number of neighbors is (3+7+8)/3 = 6.

A possible transformation: Replace each hexagon with 7 hexagons.


There exists another tetrad formed by deciamonds. Amfirifma exhibits that this deciamond can tile the plane including tetrads. Three tiles are added to the tetrad and form a shape that tiles the plane. The tetrad itself cannot tile the plane.

The yellow tiles have only 3 neighbors.