n-coloring is a way of assigning one of n different colors to each vertex of a graph such that no two adjacent vertices share the same color. It helps in determining the chromatic number of a graph, which is the smallest number of colors needed for such an assignment.
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The chromatic number of a graph is the minimum n for which an n-coloring exists.
A graph with no edges has a chromatic number of 1.
Bipartite graphs can be 2-colored, meaning their chromatic number is 2.
The Four Color Theorem states that any planar graph can be colored with at most 4 colors.
Finding the chromatic number of a general graph is an NP-hard problem.
Review Questions
What is the definition of n-coloring in the context of graph theory?
How does one determine if a given coloring is valid for an n-coloring?
What significance does the chromatic number have in relation to n-coloring?
Related terms
ChromaticNumber: The minimum number of colors required to achieve an n-coloring for a given graph.
BipartiteGraph: A type of graph that can be colored using only two colors, meaning it has a chromatic number of 2.
FourColorTheorem: A theorem stating that any planar graph can be colored with at most four different colors without two adjacent vertices sharing the same color.