Colorings with Fractional Defect

Abstract

Consider a coloring of a graph such that each vertex is assigned a fraction of each color, with the total amount of colors at each vertex summing to $1$. We define the fractional defect of a vertex $v$ to be the sum of the overlaps with each neighbor of $v$, and the fractional defect of the graph to be the maximum of the defects over all vertices. Note that this coincides with the usual definition of defect if every vertex is monochromatic. We provide results on the minimum fractional defect of $2$-colorings of some graphs.