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Vertex colouring in graphs.
Determine the minimum number of colours needed to assign a colour to each vertex of the following
five graphs, so that no two adjacent vertices get the same colour. Justify why it is the minimum
number!

The task requires determining the minimum number of colors, known as the chromatic number, needed to color each vertex of the five graphs provided, such that no two adjacent vertices share the same color. I'll analyze each graph to find its chromatic number.

### 1. **First Graph: A Cycle (C₇)**
   - This is a **7-cycle** with 7 vertices.
   - A cycle with an odd number of vertices has a chromatic number of **3**. This is because it requires alternating colors, and the odd vertex causes a need for a third color.
   - **Chromatic number**: 3.

### 2. **Second Graph: A Complete Graph (K₅)**
   - This is a **complete graph** with 5 vertices. In a complete graph, each vertex is adjacent to every other vertex.
   - A complete graph with \( n \) vertices has a chromatic number of \( n \).
   - **Chromatic number**: 5.

### 3. **Third Graph: A Ladder Graph**
   - This graph consists of two rows of 5 vertices, where each vertex is connected to its neighbor both horizontally and vertically.
   - Ladder graphs are bipartite, meaning they can be colored with 2 colors.
   - **Chromatic number**: 2.

### 4. **Fourth Graph: Bipartite with Additional Edges**
   - This graph appears to have 6 vertices arranged in a bipartite form, but with an additional edge crossing between the two partitions.
   - Since it retains its bipartite structure (two distinct sets of vertices), it can still be colored with 2 colors.
   - **Chromatic number**: 2.

### 5. **Fifth Graph: A Complete Graph with Substructure (K₅ with a Subgraph)**
   - The fifth graph resembles a **K₅** but with one internal vertex forming a pentagon-like shape, so it closely resembles a wheel graph.
   - A wheel graph with 5 outer vertices and 1 center vertex has a chromatic number of **3**.
   - **Chromatic number**: 3.

### Summary of Chromatic Numbers:
1. First graph: **3**
2. Second graph: **5**
3. Third graph: **2**
4. Fourth graph: **2**
5. Fifth graph: **3**

Would you like further explanations or a breakdown of the steps for any specific graph? Here are 5 related questions to deepen your understanding:

1. Why do odd cycles have a chromatic number of 3 while even cycles have a chromatic number of 2?
2. How does a complete graph's structure affect its chromatic number?
3. Can bipartite graphs ever require more than 2 colors for vertex coloring?
4. What are the conditions that make a graph non-planar, and how do those relate to chromatic number?
5. How does adding edges to a graph influence its chromatic number?

**Tip**: For bipartite graphs, use two sets of colors to cover vertices where no adjacent vertices are in the same set!

Vertex colouring in graphs. Determine the minimum number of colours needed to assign a colour to each vertex of the following five graphs, so that no two adjacent vertices get the same colour. Justify why it is the minimum number!

Determine the chromatic number (minimum number of colours) needed for vertex coloring in five different graphs, including cycles, complete graphs, and bipartite graphs. The solution explains key graph theory concepts and theorems used, such as the chromatic number of complete graphs and odd cycles.

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