A 
-frugal coloring is a proper
coloring of a graph in which no color appears more
than 
times among the neighbors of any vertex (Brešar
et al. 2026). A 1-frugal coloring is equivalent to a proper
coloring of the graph square.
Brešar et al. (2026) proved that every planar graph of maximum vertex degree four has a 2-frugal coloring with at most seven colors. For maximum vertex degree five, ten colors suffice.
