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The **chromatic number** of a graph is the smallest number of colors you need to color the vertices so that no two adjacent vertices share the same color. Imagine using colors to paint a map; you want regions that touch to have different colors.

In the case of our graph \( \mathcal{G} \), which is composed of a complete subgraph \( \mathcal{K}_{n-3} \) and a cycle \( \mathcal{C}_5 \), the chromatic number depends on the constraints posed by both structures.

• \( \mathcal{K}_{n-3} \): Requires \( n-3 \) different colors since it's fully connected.
• \( \mathcal{C}_5 \): A cycle with 5 vertices needs at least 3 colors, if it's odd and larger than 3, to avoid adjacent vertices sharing colors.

When these components are joined with all possible connections, the highest requirement is influenced by \( \mathcal{C}_5 \) and the connectivity combined with \( \mathcal{K}_{n-3} \). Hence, the graph's chromatic number becomes \( n \). This ensures all vertices are properly distinguished under the coloring rules, without overlap in color assignment among adjacent vertices.

A **complete graph**, denoted as \( \mathcal{K}_n \), is a simple graph where every pair of distinct vertices is connected by a unique edge. This means, if you have \( n \) vertices, then each vertex must connect to \( n-1 \) others.

For instance, \( \mathcal{K}_4 \) (complete graph with 4 vertices) would look like a fully connected square with all diagonal connections. Each vertex touches each other, with no skipped connections.

• Each vertex has a degree of \( n-1 \).
• Total number of edges is \( \frac{n(n-1)}{2} \).

In our graph structure \( \mathcal{G} \), while it uses components that are fully interconnected, it cannot form a \( \mathcal{K}_n \) as a subgraph due to the unique structure imposed by the cycle \( \mathcal{C}_5 \). This prevents full mutual connectivity among any full set of \( n \) vertices.

A **cycle graph**, noted as \( \mathcal{C}_n \), is a simple graph that forms a closed loop with \( n \) vertices and \( n \) edges. In these graphs, each vertex connects to exactly two others, forming the cycle.

Consider \( \mathcal{C}_5 \), which means 5 vertices connect in a roundabout fashion. Such cycles have a distinct property where:

• If \( n \) is odd, then at least 3 colors are needed to color the graph (as shown in a chromatic number discussion).
• If \( n \) is even, you only need 2 colors.

In the graph \( \mathcal{G} \), the presence of \( \mathcal{C}_5 \) implies unique structural properties. Its cyclic nature combined with additional connections to \( \mathcal{K}_{n-3} \) inhibits forming a complete graph \( \mathcal{K}_n \) within \( \mathcal{G} \).

A **subgraph** is essentially a smaller graph taken out from a larger one without rearranging any connections present within the graph.

For example, if you had a larger graph, you could cut out part of it, including some of the vertices and the edges connecting them, and that piece would be a subgraph.

• Subgraphs inherit edges and connections from their parent graph.
• They can be used to analyze specific parts of the graph in detail.

In the exercise, \( \mathcal{G} \) doesn't have \( \mathcal{K}_n \) as a subgraph. Even if it includes segments that are densely connected, the cycle \( \mathcal{C}_5 \) prevents forming a complete set \( \mathcal{K}_n \) because it misses some crucial links necessary for full mutual connectivity. This highlights one's need to consider how various substructures interact when analyzing graph properties.