  A 4-regular matchstick graph is a planar unit-distance graph whose vertices
have all degree 4. Examples of 4-regular matchstick graphs are currently known
for all number of vertices $\geq$ 52 except for 53, 55, 56, 58, 59, 61, and 62.
In this article we present 38 different examples with 50 - 62 vertices which
contain two, three, or four distances which differ slightly from the unit
length. These graphs should show why this subject is so extraordinarily
difficult to deal with and should also be an incentive for the interested
reader to find solutions for the missing numbers of vertices.
