https://file.aiquickdraw.com/mathbot/file/1729346663514ahjx8m21.jpg

To examine whether the given graph \( G \) is bipartite, let's proceed step by step.

### **Definition of a Bipartite Graph:**
A graph is **bipartite** if its set of vertices can be divided into two distinct sets, say \( U \) and \( V \), such that:
1. Every edge connects a vertex in \( U \) to a vertex in \( V \), and
2. No edge exists between vertices within the same set.

A useful method for determining whether a graph is bipartite is to **attempt to 2-color** the graph (i.e., assign two colors to the vertices such that no two adjacent vertices have the same color). If this can be done, the graph is bipartite.

### **Step 1: Label the Vertices**
From the image, the vertices are labeled as \( a, b, c, d, e, f, g \).

### **Step 2: Check for Odd-Length Cycles**
A key property of bipartite graphs is that they **do not contain odd-length cycles**. If an odd-length cycle exists, the graph cannot be bipartite.

#### Observation:
- Check the cycle \( a \rightarrow b \rightarrow d \rightarrow e \rightarrow a \).
  This is a cycle of length 4, which is even, so it does not disqualify the graph.

- Another cycle: \( a \rightarrow f \rightarrow g \rightarrow a \). This is a triangle (3-cycle), which is of odd length.

Since a triangle (3-cycle) exists in the graph, it implies that the graph is **not bipartite**.

### **Conclusion:**
The graph \( G \) is **not bipartite** because it contains a cycle of odd length (3-cycle).

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Would you like further details or have any questions? 

Here are 5 related questions to deepen your understanding:
1. What are the main differences between bipartite and non-bipartite graphs?
2. How can you algorithmically check if any graph is bipartite?
3. What are real-life applications of bipartite graphs?
4. Can bipartite graphs have cycles at all, and if so, what kind?
5. How does the absence of odd-length cycles guarantee bipartiteness?

**Tip:** A quick way to check if a graph is bipartite is to use breadth-first search (BFS) and try coloring the graph with two colors.

Examine whether the given graph G is bipartite.

Examine whether the given graph G is bipartite. Follow a step-by-step approach using graph theory concepts like bipartite graphs and odd-length cycles. This example includes definitions and identification of cycles to explain why the graph is not bipartite. Suitable for students in Grades 10-12.

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