The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least Delta + 1 colors, where Delta is the maximum degree in the graph. For all Delta, Wegner constructed planar graphs (even without 3-cycles) that require about rac{3}{2} Delta colors for such a coloring.To prove a stronger upper bound, we consider only planar graphs that contain no 4-cycles and no 5-cycles (but which may contain 3-cycles). Zhu, Lu, Wang, and Chen showed that for a graph G in this class with Delta ge 9, we can color G^2 using no more than Delta + 5 colors. In this thesis we improve this result, showing that for a planar graph G with maximum degree Delta ge 32 having no 4-cycles and no 5-cycles, at most Delta + 3 colors are needed to properly color G^2. Our approach uses the discharging method, and the result extends to list-coloring and other related coloring concepts as well.
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