To show that a connected graph $G$ is bipartite if and only if it has no odd cycles, we will prove both directions of the equivalence.

Proof

(⇒) If $G$ is bipartite, then $G$ has no odd cycles.

1. Definition of Bipartite Graph: A graph is bipartite if its vertex set can be divided into two disjoint sets $U$ and $V$ such that every edge in the graph connects a vertex in $U$ to a vertex in $V$.

2. Cycle Properties: Consider any cycle in $G$. Since $G$ is bipartite, the vertices of the cycle alternate between the two sets $U$ and $V$.

3. Cycle Length: If a cycle has an odd number of vertices, say $k$, then starting from a vertex in $U$, the alternating pattern means that the vertices must return to the starting vertex after $k$ steps. This results in the last vertex being in the same set as the starting vertex, contradicting the bipartite nature.

4. Conclusion: Therefore, if $G$ is bipartite, it cannot contain any odd-length cycles.

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(⇐) If $G$ has no odd cycles, then $G$ is bipartite.

1. Coloring the Graph: We will use a coloring argument. Start by choosing any vertex $v$ in $G$ and color it with color 1 (say red).

2. Breadth-First Search (BFS): Perform a BFS from vertex $v$. Color all its adjacent vertices with color 2 (say blue).

3. Continuing the BFS: For every vertex colored red, color its unvisited neighbors blue, and for every vertex colored blue, color its unvisited neighbors red. Continue this process until all vertices are colored.

4. Checking for Odd Cycles: Since we assumed $G$ has no odd cycles, any cycle encountered during this coloring process must be of even length. Thus, the alternating colors confirm that adjacent vertices are always of different colors.

5. Conclusion: Since we can color $G$ in this way, $G$ is bipartite.

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Final Conclusion

We have shown both directions of the implication. Thus, a connected graph $G$ is bipartite if and only if it has no odd cycles.

Questions for Further Exploration

1. What is an example of a bipartite graph with cycles?
2. How can we efficiently check if a graph is bipartite?
3. What are the implications of bipartiteness in network flow problems?
4. Can a graph have both odd and even cycles? What would that imply about its bipartiteness?
5. How does the concept of bipartite graphs apply to real-world problems, such as matching jobs to applicants?

Tip

When proving properties of graphs, visualizing the structure through drawing can greatly aid in understanding the relationships between vertices and edges.