Prove if the following statement is false or true: If G is not bipartite, then the shortest cycle in G has odd length.
Prove if the following statement is false or true: If G is not bipartite, then the shortest cycle in G has odd length.
Introduction
Bipartite:
A graph is called bipartite if its vertices can be divided into two disjoint sets, let's call them U and V, such that every edge in the graph connects a vertex in U to a vertex in V (i.e., every edge has one endpoint in U and one endpoint in V). Another way to put it is that the graph can be colored with two colors, such that no two vertices of the same color are adjacent.
In other words, a bipartite graph is a graph that does not contain any odd-length cycles. This is because if a graph contains an odd-length cycle, then it is impossible to color the vertices of the cycle with only two colors without coloring adjacent vertices with the same color.
Bipartite graphs have many interesting properties and applications. For example, they can be used to model relationships between different types of objects, such as customers and products, or students and courses. They also have efficient algorithms for various problems, such as maximum matching, which finds the largest possible set of edges that do not share endpoints in a bipartite graph.
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