In multi-cell massive multiple-input multiple-output (MIMO) cellular networks, pilot contamination (PC) caused by ineluctable pilot reuse severely restricts the system performance in the effectiveness and reliability of transmission. To deal with this problem, an efficient pilot assignment scheme based on fractional pilot reuse and max k-cut (FPR-MKC) is therefore proposed in this study. Specifically, first, the measure of susceptibility to interference for each user is designed, and based on this, an innovative boundary is determined for properly selecting the edge users who could employ the unique pilot sequences. Subsequently, the process of allocating reused pilots is innovatively treated as the partition of the vertices of a graph. Building on this idea, an edge-weighted undirected graph is constructed to present the potential interference intensity among users, and eventually, the max k-cut (MKC) strategy is executed, which aims to partition centre users into a desired number of mutually exclusive subsets with maximizing the total weight of the edges between the disjoint subsets. Compared with some existing pilot assignment schemes, the proposed FPR-MKC scheme can adequately mitigate the PC with significantly enhanced quality of service for edge users, which is verified by theoretical analysis and numerical simulation results. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.