<![CDATA[Suppose is a graph and is a subgraph of . All edges of graph are given arbitrary colors such that there is a properly colored subgaph . However, this problem is difficult to prove, thus the problem is given a constraint. Suppose is a complete graph with vertices with 4 as the minimum for , and (subgraph of ) is a cycle with 4 vertices. All edges of complete graph G are given arbitrary colors such that there is a properly colored subgraph cycle . This study has proven that there is a sufficient condition that graph contains a properly colored cycle through the minimum color degree. This study has also proven that there is a sufficient condition that graph contains a properly colored cycle through the cardinality of the union of the neighboring color of vertex and neighboring color of vertex . Keywords: Graph coloring, complete graph, cycle graph, as well as minimum color degree.]]>
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