The spanning diameter of the star graphs

Abstract

Assume that u and v are any two distinct vertices of different partite sets of S/sub n/ with n /spl ges/ 5. We prove that there are (n - 1) internally disjoint paths P/sub 1/, P/sub 2/, ..., P/sub n-i/ joining u to v such that /spl cup//sup n = 1//sub i = 2/ P/sub i/ spans S/sub n/ and l(P/sub i/) /spl les/ (n - 1)! + 2(n - 2)! + 2(n - 3)! + 1 = n!/(n - 2) + 1. We also prove that there are two internally disjoint paths Q/sub 1/ and Q/sub 2/ joining u to v such that Q/sub 1/ /spl cup/ Q/sub 2/ spans S/sub n/ and l(Q/sub i/) /spl les/ n!/2 + l for i = 1,2.