几类和扇有关图的优美性

证明下面的结论:对任意自然数n≥2,图(K_1∨(P_n∪P_(n+1)))是(n-1)-强优美图.对任意自然数n≥3,图(K_1∨P_n~((1))∪P_n~((2))))∪G是优美图;对任意自然数n≥4,图(K _1∨(P_n~((1))∪P_n~((2))∪P_n~((3)))∪H是优美图,其中k=n/2].P_n是n个顶点的路,G_i为含有i条边的优美图.给定优美图G_(n-1)和其优美标号f,G_(k-1)和其优美标号g,设u∈G_(n-1),v∈G_(k-1)且f(u)=g(v)=0,取不同的两边xy和x′y′,点x与u合并后得到的图记为G,点x′与v合并后得到的图记为H.

Gracefulness of some graphs related to Fan

This paper contained the following results :for any natural number n≥2 ,the graph K1 ∨ Pn ∪ Pn+1 was (n-1)‐strong graceful ;for any natural number n≥3 ,the graph K 1 ∨ P(1 )n ∪ P(2 )n∪ G was graceful ; for any natural number n ≥ 4 , the graph K1 ∨ P(1)n ∪ P(2)n ∪ P(3)n ∪ H was graceful ,where k= n2 ,Pn be a path with n vertices , and Gi be a graceful graph with i edges .Given a graceful graph Gn-1 with its graceful labeling f ,and Gk-1 with its graceful labeling g ,we assumed that a vertex u∈ Gn-1 ,v∈ Gk-1 with f(u)= g(v)=0 .Taking two copies of P2 ,xy and x′y′,identifying vertices x and u ,x′and v ,we obtained the resulting graph G and H respectively .