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While I know nothing about these measures, quick googling led me to the paper Michael O. Albertson, Glenn G. Chappell, H. A. Kierstead, André Kündgen, Radhika Ramamurthi: Coloring with no $2$-colored $P_4$’s, Electronic Journal of Combinatorics 11 (2004), no. 1, art. no. R26, doi: 10.37236/1779. They prove that $\chi_s(G)\le\chi_a(G)\bigl(2\chi_a(G)-1\bigr)$ ...


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