1-planar graphs were first studied by Ringel (1965), who showed that they can be colored with at most seven colors. Later, the precise number of colors needed to color these graphs, in the worst case, was shown to be six.
I have constructed a large class of 1-planar graphs with connectivity 6 and chromatic number 6. However, when the connectivity increases to 7, so far, I have not seen a 1-planar graph with connectivity 7 that has chromatic number exactly equal to 6. Even when considering 1-planar graphs with minimum degree 7, I have not found such examples. Note that there is a large class of 7-connected 1-planar graphs with chromatic numbers of 4 or 5.
Thus, the following question puzzles me: Is it possible to construct a 1-planar graph with connectivity 7 or minimum degree 7 that has chromatic number 6?
