Discussion: In this second example of this problem this student takes the problem further on their own volition: As in the first example it is shown using the pigeonhole principle that if there is no isolated vertex then there are two vertices with the same degree. However, this student notices that if there is an isolated vertex then none of the remaining vertices has degree n-1 and so the possibilities for the remaining n -1 vertices are 0,1,2, . . ., n-2. If one of these is isolated then there are two vertices with degree 0, otherwise the possibilities are 1,2, . . ., n-2 for n-1 vertices and so two have the same degree.