The Gracefulness of Unconnected Graphs(P_3∨■)∪G and(C_3∨■)∪G
Deming Li · Acta Scientiarum Naturalium Universitatis Sunyatseni · 2012
The definition of k-graceful graph is extended and the new concept of A~B graceful graph is presented.One sufficient condition about determining the gracefulness of unconnected graph(P3 ∨■) ∪ G and(C3 ∨ ■) ∪ G is obtained.In the meanwhile,it is proved that for any natural numbers k,m,n,t,which are not less than one,when k≤n≤t and n+k-1≤m,the disconnected graphs(P3 ∨Km)∪(∪k j=1Kn,t) and(C3 ∨■)∪(∪k j=1Kn,t) are graceful;when k=1,2,2≤n2m+1 the graphs(P3 ∨■)∪∪k j=1Pn,(C3 ∨■)∪∪k j=1Pn and(P3 ∨■)∪Pn∪St(t) are graceful;when 2≤n≤2m+1,the graphs(C3 ∨■)∪Pn∪St(t) are graceful.Results generalize some of the known results.