Three multiply connected Kn-residual graphs

The $ {K_n}\textrm{ - } $ residual graph was proposed by P. Erdös, F. Harary and M. Klaw. They also proposed conclusions and conjectures regarding connected $ m - {K_n}\textrm{ - } $ residual graphs. When $ m = 1 $ , $ n \ne 1,2,3,4 $ , the authors proved that $ {K_{n + 1}} \times {K_2} $ is the only connected residual graph with a minimum order. In this paper, we proved that there are three different connected $ 3 - {K_5}\textrm{ - } $ residual graphs with a minimum order of 32, a unique connected $ 3 - {K_6}\textrm{ - } $ residual graph with a minimum order of 33, and a unique connected $ 3 - {K_8}\textrm{ - } $ residual graph with a minimum order of 44, which is not isomorphic to $ {K_{11}} \times {K_4} $ . At the same time, when $ n \ge 5,n \ne 6 $ , we proved that the minimum order of a connected $ 3 - {K_n}\textrm{ - } $ residual graph is $ 4n + 12 $ , and when $ n \ge 7,n \ne 8, $ $ {K_{n + 3}} \times {K_4} $ is the unique smallest connected $ 3 - {K_n}\textrm{ - } $ residual graph. Therefore, we verified the conjecture about connected $ 3 - {K_n}\textrm{ - } $ residual graphs. When $ n \ge 5 $ , we could obtain the minimum order and specify the corresponding extremal graph of the connected $ 3 - {K_n}\textrm{ - } $ residual graphs.