In a standard graph-coloring problem, it is required that all edges will be bichromatic (i.e., all edges should be connected to two vertices with different colors). What is a term, and some basic references, for a graph-coloring problem in which only a fraction of the edges should satisfy this condition? For example, at least $\frac23$ or at least $\frac12$?
I looked for "fractional coloring" but, apparently, it is a completely different problem.
A colouring without the condition that edges be bicromatic is called by different names such as relaxed coloring, improper coloring and defective coloring. (warning: the term defective coloring is more commonly used to mean a relaxed coloring with a restriction on maximum degree allowed in colour classes; relaxed coloring with this restriction is explored enough in the literature in both theoretical interest and applications)
What you want is a relaxed coloring with a fraction of edges coloured properly. I remember seeing something similar to this on a paper on .... defective colouring(?), but quite don't remember the source.
I thought this could help someone interested to further explore.