Bipartite Graphs: Does $\alpha(G) =|U|$?

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In summary, a bipartite graph is a graph with two distinct sets of vertices, where all edges connect vertices from one set to the other set. The notation $\alpha(G) = |U|$ refers to the size of the maximum independent set in a bipartite graph, which is equal to the number of vertices in one of the two sets of vertices. This is a property specific to bipartite graphs and is always true. This equality can be useful in various applications, such as scheduling or matching problems, as it represents the maximum number of vertices that can be selected without any edges between them.
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Let $G$ be a bipartite graph with bipartition $U$ and $W$ such that $|U|\ge |W|$. Is it true that $\alpha(G) =|U|$?The answer is false, but I don't know how to justify it. I would appreciate any help.
 
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If you do not assume connectedness then the simplest example where $\alpha(G)$ is more than $|U|$ is when there are no edges. But one can constrcut such exampels even with connectedness.
 

FAQ: Bipartite Graphs: Does $\alpha(G) =|U|$?

What is a bipartite graph?

A bipartite graph is a type of graph where the vertices can be divided into two disjoint sets, such that all edges only connect vertices from different sets. This means that there are no edges between vertices within the same set.

What is the significance of $\alpha(G) =|U|$ in bipartite graphs?

In bipartite graphs, $\alpha(G) =|U|$ means that the maximum independent set (a set of vertices where no two are adjacent) in the graph is equal to the size of one of the vertex sets. This is a special property of bipartite graphs and can be used to solve certain problems.

How can we determine if a graph is bipartite?

A graph can be determined to be bipartite by using a graph coloring algorithm. If the graph can be colored with two colors such that no adjacent vertices have the same color, then it is bipartite. Alternatively, if the graph has no odd cycles, it is also bipartite.

Can a bipartite graph have an odd number of vertices?

Yes, a bipartite graph can have an odd number of vertices. The number of vertices does not affect the property of being bipartite, as long as the graph can be divided into two disjoint sets of vertices.

Are there any real-world applications of bipartite graphs?

Yes, bipartite graphs have many real-world applications. They are commonly used in matching problems, such as matching students to schools or employees to job positions. They are also used in recommendation systems, where one set of vertices represents users and the other represents items, and the edges represent ratings or preferences. Bipartite graphs also have applications in genetics, social networks, and transportation networks.

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