Question

Prove that all bipartite graphs are perfect.

   Prove that all bipartite graphs are perfect.

Introductory Combinatorics

Introductory Combinatorics

Richard A. Brualdi 5th Edition

Chapter 12, Problem 45

Instant Answer

Solved by Expert Wen Zheng

09/07/2019

Step 1

First, let's recall the definition of a bipartite graph. A graph G is bipartite if its vertex set can be partitioned into two disjoint sets U and V such that every edge in G connects a vertex in U to a vertex in V. Show more…

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Prove that all bipartite graphs are perfect.

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Key Concepts

Bipartite Graphs

A bipartite graph is one whose vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other. This property guarantees that the graph contains no odd-length cycles and can be colored with just two colors, which is a fundamental aspect when considering its chromatic properties.

Perfect Graphs

A perfect graph is defined by the property that for every induced subgraph, the chromatic number (the minimum number of colors needed to color the vertices so that no two adjacent vertices share the same color) is equal to the clique number (the size of the largest complete subgraph). This means that the graph and all of its induced subgraphs can be optimally colored using as many colors as the size of their largest cliques.

Chromatic Number

The chromatic number of a graph is the minimum number of colors required to assign to its vertices such that no two adjacent vertices share the same color. For bipartite graphs, this number is at most 2, which aligns with their structure of having two independent sets.

Clique Number

The clique number of a graph is the size of the largest set of vertices that forms a complete subgraph (where every pair of vertices is connected by an edge). In bipartite graphs, the largest possible clique can have at most two vertices, since any larger clique would violate the bipartite property.

Induced Subgraph

An induced subgraph is formed by selecting a subset of vertices from the original graph and including all the edges between them that exist in the original graph. The concept is crucial for the definition of perfect graphs, as perfectness requires that every induced subgraph also satisfies the property that its chromatic number equals its clique number.

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