Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Definition: A coloring of a simple graph is the assignment of a color to each
vertex of the graph so that no adjacent vertices are assigned the same color.
Definition: The chromomatic number F(G) is the least number of colors needed
for a coloring of the graph G = (V,E).
Definition: A planar graph is a graph that can be drawn in a plane with no edges
crossing in the picture.
Four Color Theorem: If G is a planar graph, then F(G) ; 4.
Note: The Four Color Conjecture was made in the 1850’s and not proven until
1976. Like Fermat’s last theorem, this theorem became famous partly for how
many wrong proofs (some quite ingenious) were either published or submitted
for publication.
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Trees
Definition: A tree is a connected undirected graph with no simple circuits. A
weighted tree is a tree with weights associated with the edges.
Uses:
• An efficient data structure for searching a list.
o Useful in encoding data for transmission.
o Computational complexity easily determined for algorithms using trees.
• Weighted trees have edges with weights.
o Useful in decision making.
o Used by telecoms to dynamically connect calls cheaply.
Historical Note: Trees were first developed in the context of this course to
describe molecules in chemistry, where atoms were the vertices and bonds were
the edges.
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