What are Fractional Colorings? (Graph Theory Tutorial)
What is fractional coloring? This video defines fractional coloring and explains its important properties.
Typically, when we talk about graph colorings, we mean assigning a color to each vertex in the graph such that adjacent vertices are of different colors. However, in fractional colorings, we assign a set of colors to each vertex in the graph, such that the color-sets of adjacent vertices are disjoint, or share no colors in common. In other words, we assign multiple colors to each vertex, and if two vertices are connected, they have no colors in common.
If we assign sets of 2 colors to each vertex in the graph, that is called a 2-fold coloring. If we assign sets of 3 colors to each vertex in the graph, that is called a 3-fold coloring. An n-fold coloring of a graph is one in which we assign sets of size n to each vertex in the graph, such that adjacent vertices are assigned disjoint sets of colors. Using b-fold coloring, we can define the fractional chromatic number of a graph.
Fractional coloring is highly applicable to real world problems as well. It's used in activity scheduling problems, which can come up in the operation of a wireless communication network.
If you found fractional colorings interesting, I'd encourage you to look into the branch of math called fractional graph theory. Here is a good book about it: "Fractional Graph Theory" by Scheinerman and Ullman.
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