In fact, there are examples of graphs for which \(\chi(G) = \Delta(G) + 1\text{.}\) For any \(n\text{,}\) the complete graph \(K_n\) has chromatic number \(n\text{,}\) but \(\Delta(K_n) = n-1\) (since every vertex is adjacent to every other vertex). Additionally, any odd cycle will have chromatic number 3, but the degree of every vertex in a cycle is 2. It turns out that these are the only two types of examples where we get equality, a result known as Brooks' Theorem.
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