This paper gives some new results on mutually orthogonal graph squares (MOGS). These generalize mutually orthogonal Latin squares in an interesting way. As such, the topic is quite nice and should have broad appeal. MOGS have strong connections to core fields of finite algebra, cryptography, finite geometry, and design of experiments. We are concerned with the mutually orthogonal half starters method to construct the mutually orthogonal graph squares for disjoint union of paths.

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