Prove that every simple 9-regular graph on 100 vertices contains a subgraph with maximum degree at most 5 and at least 225 edges.
My attempt
I think I should use the pigeonhole principle but I don't know how.
I thought maybe to take 2 subgraphs that complete each other to the whole graph and in one of them the maximum degree has to be 5 (if in both of them it's above 5 than the maximum degree of the whole graph is above 10). but how do I know that this subgraph has also at least 225 edges?
I'm not sure that is necessarily true. I'd like to get some help here, thank you!
By Vizing’s theorem there exists a proper edge-coloring of the graph in 10 colors. So there exist 5 colors coloring at least $(100\cdot 9/2)\cdot 5/10=225$ edges. The graph with these edges has maximal vertex degree at most 5.