  Proper graph coloring assigns different colors to adjacent vertices of the
graph. Usually, the number of colors is fixed or as small as possible. Consider
applications (e.g. variants of scheduling) where colors represent limited
resources and graph represents conflicts, i.e., two adjacent vertices cannot
obtain the same resource. In such applications, it is common that some vertices
have preferred resource(s). However, unfortunately, it is not usually possible
to satisfy all such preferences. The notion called flexibility was recently
defined in [Dvo\v{r}\'ak, Norin, Postle: List coloring with requests, Journal
of Graph Theory 2019]. There instead of satisfying all the preferences the aim
is to satisfy at least a constant fraction of the request. Recently, the
structural properties of planar graphs in terms of flexibility were
investigated. We continue this line of research. Let G be a planar graph with a
list assignment L. Suppose a preferred color is given for some of the vertices.
We prove that if G is a planar graph without 4-cycles and all lists have size
at least five, then there exists an L-coloring respecting at least a constant
fraction of the preferences.
