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exanded answer
Aaron Meyerowitz
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Maximum degree 2 would be such a class (which includes regular of degree $2$ as a subclass). Transitive graphs (by which I mean that the relation of being connected by an edge is transitive) are another example (there is a less obscure description of that class of graphs but I wanted it to sound mysterious for a few moments).

Aaron Meyerowitz
  • 30.1k
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  • 104