[kanji]

แนะก็ได้นะครับ ขอบคุณครับ
1.Prove that the Petersen graph has no cycle of length 7.
2.Decompose the Petersen graph into three connected subgraphs that are pairwise isomorphic.Also decompose it into copies of $P_4$.
3.Prove that $K_n$ decomposes into three pairwise-isomorphic subgraphs if and only if $n+1$ is not divisible by 3.
(Hint:For the case where $n$ is divisible by 3,split the vertices into three sets of equal size.)
4.Let $G$ be a simple graph in which every vertex has degree 3.
Prove that $G$ decomposes into claws if and only if $G$ is bipartite.
5.Construct a simple graph with six vertices that has only one automorphism.
Construct a simple graph that has exactly three automorphisms.
(Hint:Think of a rotating triangle with appendages to prevent flips.)