Solving the dynamic coloring problem for direct products of paths with fan graphs

This paper deals with the r-dynamic chromatic problem of the direct product of a path with a fan graph F_m,n. The problem is completely solved except for the case n<r ∈{2m +2, 2m +3}, which is solved under certain assumptions. It enables us to determine in particular the dynamic chromatic number concerning this problem, for all r ≤ 7, and also, for all m ∈{1, 2}.