Question

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is ............ \begin{center} \includegraphics[width=0.3\textwidth]{60.png} \end{center}

Hint:

Bipartite graphs always have a chromatic number of \(2\), as their vertices can be divided into two independent sets.

Answer 1

Step 1: Definition of chromatic number. The chromatic number of a graph is the smallest number of colours required to colour its vertices such that no two adjacent vertices share the same colour. Step 2: Analyze the graph structure. The given graph is a bipartite graph, as the vertices can be divided into two disjoint sets such that: All edges connect vertices from one set to the other. There are no edges between vertices within the same set. A bipartite graph can always be coloured using exactly \(2\) colours. Step 3: Verify bipartite property. The graph has two disjoint sets of vertices: \[ \text{Set 1: Outer vertices.} \quad \text{Set 2: Inner vertices.} \] All edges connect vertices from Set 1 to Set 2, satisfying the bipartite condition. Thus, the chromatic number of the graph is \(2\). Final Answer: \[ \boxed{2} \]