完备二分图的冠的k—优美性

The K—Gracefulness of the Cornonal of Complete Bipartite Graphs

图的优美性是图的重要研究内容之一,有广泛的应用背景.1991年,马克杰提出猜想:完备二分图km,n的冠I(km,n)是k—优美图,其中m,n,k是任意正整数且m≤n.当m=2,3,4,5或k>(m-2)n;m=1,2;或k≥(m-2)(n-1)的情形,在文献〔6,7〕中证明了猜想的正确性.本文利用构造方法也给出了对于任意正整数m,n,k时,当m<n,m≥6,n≥m2-3m+2/2时,完备二分图km,n的冠I(km,n)的另一种k—优美标号.

The gracefulness of graphs which having widely applications is an important property of graphs. Makejie guessed that the coronal I（Km,n） of any complete bipartite graph Km,n is a K-graceful graph,where m,n,k are any positive integers and m ≤ n. In this paper, we proof the coronal I（Km,n ） of any complete 2 bipartite graph Km,n is a K-graceful Graph when m ≥6,n ≥（m^2-3m＋2）/2 m〈n and m,n,k are any positive integers. Furthermore,the coronal I（Km,n ） of any complete bipartite graph Km,n is not K-graceful graph. The result enriches the graceful theory of graphs.