非连通图(P1∨Pn)∪Gr和(P1∨Pn)∪(P3∨r)及Wn∪St(m)的优美性

讨论非连通图((P₁∨P_n)∪G_r和(P₁∨P_n)∪(P₃∨_r)及W_n∪St(m)的优美性, 证明了如下结论： 设n,m为任意正整数, s=［n/2］, r=s-1, G_r是任意具有r条边的优美图, 则当n≥4时, 非连通图((P₁∨P_n)∪G_r和(P₁∨P_n)∪(P₃∨_r)是优美图； 当n≥3, m≥s时, 非连通图W_n∪St(m)是优美图. 其中, P_n是n个顶点的路, K_n是n个顶点的完全图, n是K_n的补图, G₁∨G₂是图G₁与G₂的联图, W_n是n+1个顶点的轮图, St(m)是m+1个顶点的星形树.

Gracefulness of Unconnected Graphs (P1∨Pn)∪Gr,(P1∨Pn)∪(P3∨(K)r) and Wn∪St(m)

The present paper deals with the gracefulness of three kinds of unconnected graphs ((P₁∨P_n)∪G_r,(P₁∨P_n)∪(P₃∨_r) and W_n∪St(m), and proves the following results： for positive integers n and m, let s=［n/2］, r=s-1, G_r be a graceful graph with redges, if n≥4, then the unconnected graphs (P₁∨P_n)∪G_r,(P₁∨P_n)∪(P₃∨_r) are both graceful graphs； if n≥3andm≥s, then the unconnected graph W_n∪St(m) is a graceful graph, where P_n is an nvertex path, K_n is an n vertex complete graph, nis the complementof graph K_n, graph G₁∨G₂ is the join graph of G₁ and G₂, W_nis an (n+1)vertex wheel graph and St(m)is an (m+1) vertex star tree.