# Graphs of maximum degree three

I'm learning an algorithm for graphs of maximum degree three.

My question is: should the graph of that type have at least one vertex with degree three.

For example if the maximum degree of some graph is 2, can we say that this graph is maximum degree three.

Thank you

• It ultimately depends on the definition you use, but usually a "maximum" does not require a sharp bound. May 23, 2019 at 12:05
• Usually, "maximum degree 3" is really "maximum degree at most 3", i.e., "all vertices have degree at most 3". You could say "maximum degree exactly 3" if you want the maximum degree to be exactly 3. But the exact meaning really depends on the context. May 23, 2019 at 13:20

Similarly, a function of degree $$d$$ sometimes means a function of degree exactly $$d$$, and sometimes a function of degree at most $$d$$. In many contexts, the difference is immaterial. For example, a graph of maximum degree 3 on $$n$$ vertices has at most $$1.5n$$ edges – and this holds for both meanings.