5. (8 pts.) Let n be a positive integer. Prove that if the complete graph K, can be de- composed into triangles (i.e., there is a set of subgraphs of K, such that each subgraph is isomorphic then n - 1 or n 3 is divisible by 6. Find an explicit decomposition to triangles of K7 and K9 (number the vertices, and provide the triples of the decompositions) (6 pts.) Show that every loopless graph C has a bipartite spenning subgraph wit the property that deso for every vertes eVG) to K3, and each edge of Kn appears in exactly one of the triangles), that degy(v) > ld o nhlame 1-4 and the following 5. (8 pts.) Let n be a positive integer. Prove that if the complete graph K, can be de- composed into triangles (i.e., there is a set of subgraphs of K, such that each subgraph is isomorphic then n - 1 or n 3 is divisible by 6. Find an explicit decomposition to triangles of K7 and K9 (number the vertices, and provide the triples of the decompositions) (6 pts.) Show that every loopless graph C has a bipartite spenning subgraph wit the property that deso for every vertes eVG) to K3, and each edge of Kn appears in exactly one of the triangles), that degy(v) > ld o nhlame 1-4 and the following