Complex sentence: In a complete graph, the only distinguishing colorings assign a different color to each vertex because there would exist a symmetry that swapped two vertices if two vertices were assigned the same color, leaving the rest in place.
Split into simpler sentences: In a complete graph, the only distinguishing colorings assign a different color to each vertex. For, if two vertices were assigned the same color, there would exist a symmetry that swapped those two vertices, leaving the rest in place.