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Triangle counting (TC) is a fundamental problem in graph analysis and has found numerous applications, which motivates many TC acceleration solutions in the traditional computing platforms like GPU and FPGA. However, these approaches suffer from the bandwidth bottleneck because TC calculation involves a large amount of data transfers. In this paper, we propose to overcome this challenge by designing a TC accelerator utilizing the emerging processing-in-MRAM (PIM) architecture. The true innovation behind our approach is a novel method to perform TC with bitwise logic operations (such as \texttt{AND}), instead of the traditional approaches such as matrix computations. This enables the efficient in-memory implementations of TC computation, which we demonstrate in this paper with computational Spin-Transfer Torque Magnetic RAM (STT-MRAM) arrays. Furthermore, we develop customized graph slicing and mapping techniques to speed up the computation and reduce the energy consumption. We use a device-to-architecture co-simulation framework to validate our proposed TC accelerator. The results show that our data mapping strategy could reduce $99.99\%$ of the computation and $72\%$ of the memory \texttt{WRITE} operations. Compared with the existing GPU or FPGA accelerators, our in-memory accelerator achieves speedups of $9\times$ and $23.4\times$, respectively, and a $20.6\times$ energy efficiency improvement over the FPGA accelerator.