Colorings with Fractional Defect

TL;DR

This paper explores fractional colorings of graphs, defining a fractional defect measure, and investigates the minimum possible defect for 2-colorings across various graphs, extending traditional defect concepts.

Contribution

It introduces the fractional defect concept for graph colorings and analyzes the minimal defect achievable with 2-colorings in different graph classes.

Findings

Determined bounds for fractional defect in specific graphs

Extended defect analysis from monochromatic to fractional colorings

Provided methods to minimize fractional defect in 2-colorings

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.