I tried to make the degrees of vertices small since the bigger the degrees get, the more possible this cycle occurs. Hence, I first drew $14$-sided polygon so now we have a graph with $14$ vertices and $14$ edges. Then I linked every vertices with itself so we have only $1$ edge left. However, if I either put this last edge between two adjacent vertices or connect one vertex with itself again, it seems that this cycle does not occur. Does it? Where is the mistake? Is there another way to approach this question or is my reasoning true?