The 12-vertex, 26-edge graph shown above is not 3-colorable, and is 4-colorable in 178 ways, up to permutation of the four colors. Numerical evidence suggests that it is a unit distance graph, i.e. embeddable in the plane with all edges of unit length.
My unit distance graph solver found no 12-vertex, non 3-colorable, 4-colorable udg with less than 178 different colorings, counting permutations of the four colors as equivalent. The search was not exahaustive and lasted a day.
Below, I copy the adjacency matrix for the graph above. The rows and columns are not labeled, but it is easy to guess: A for row 1, B for row 2, C for row 3, …. L for row 12; similarly for the labeling of the columns.
I don’t know how to do a fixed-width, or monospace font, in WordPress, so I omit the labels to avoid misalignment.
Adjacency matrix, 12 by 12:
0 1 0 0 0 1 1 1 1 0 0 0
1 0 0 0 0 0 0 0 1 0 1 1
0 0 0 1 0 1 1 0 0 1 0 1
0 0 1 0 1 1 0 0 1 0 0 0
0 0 0 1 0 0 0 1 1 1 1 0
1 0 1 1 0 0 0 1 0 0 0 0
1 0 1 0 0 0 0 0 1 1 0 0
1 0 0 0 1 1 0 0 0 1 1 0
1 1 0 1 1 0 1 0 0 0 0 0
0 0 1 0 1 0 1 1 0 0 0 1
0 1 0 0 1 0 0 1 0 0 0 0
0 1 1 0 0 0 0 0 0 1 0 0
