The problem reads: "Show that if a graph G contains a closed trail of length 7, then G contains  C_3 , C_5 , or  C_7 as a subgraph. Can something similar be stated if G contains a closed trail of length 8?"

To begin the problem, I drew a graph that looks like a hexagon with a loop on one of the vertices. But I'm not understanding where the smaller cycle sizes come in. I'm not sure what I'm doing wrong.