k-ary n立方体中的测地泛圈

文中用归纳假设法证明了结论：当n≥2,k≥3,u和v是Qkn中任意2个顶点,由对称性,不妨设u=（0,0,…,0）,v=（d1,d2,…,dn）,这里0≤di≤k/2,（i=1,…,n）,记d=d1＋d2＋…＋dn≤1,N=kn,则对于每个偶数l适合2d＋2≤l≤N,则Qkn中有过u和v长为l的圈C,且C上u和v的距离为d.若有i和j满足1≤i≤j≤n,使得di≥1且dj≥1,或有且dj=k/2且dj=0,j≠i,1≤j≤n,则又有l=2d;当n≥2,k≥3是奇数,u和v是Qkn中任意2个顶点,由对称性,不妨设u=（0,0,…,0）,v=（d1,d2,…,dn）,这里0≤di≤k/2,,（i=1,…,n）,记d=d1＋d2＋…＋dn≥1,N=kn,r=max{di},则对于每个奇数l适合2d＋k-2r≤l≤N,则Qkn中有过u和v长为l的圈C,且C上u和v的距离为d.

Geodesic Pancyclicity of k-ary n-cube

In this paper, the following result is obtained. Let Qkn be the k-ary n-cube, where n≥2,k≥3, Assume that u=（0,0,…,0） and v=（d1, d2,…,dn） be distinct vertices of Qkn, d =d1＋ d2＋ …＋ dn≥1 and N=kn where 0≤di≤k/2,（i=1,…,n）, there exists a cycle C of every even length from 2d＋ 2 to N and the distance of u and v in C is d in Qkn. Moreover,if 1≤i≤j≤n such di≥1 and dj≥1 or and di= k/2 and di=0 where 1≤j≤n, j≠i, then there exists a cycle C of every even length of 2d in Qkn. Let Qkn be the k-ary n-cube, where n≥2, k≥3 and k is odd, Assume that u and v be distinct vertices of Qkn, Let u=（0,0,…,0）, v=（d1, d2,…,dn）, d =d1＋ d2＋ …＋ dn≥1, N=knand r=max{di}, where 0≤di≤k/2,（i=1,…,n）,there exists a cycle C of every odd length from 2d＋ k-2r to N and the distance of u and v in C is d in Qkn.