Unbounded planar skewness–excess gap for circulant graphs

Determine whether the difference between planar skewness and planar Euler excess is unbounded for the circulant graphs c(3k+1,k) = C(3k+1,{1,k}) as k increases.

Background

The discussion contrasts planar excess and crossing number for circulant graphs. For the family c(3k+1,k), a cited result shows that the planar crossing number grows with k, whereas the planar excess remains fixed at four for the relevant triangle-free 4-regular graphs.

The paper notes that this yields an arbitrarily large crossing-number–excess gap, but leaves unresolved whether the analogous gap involving skewness is also unbounded.