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This paper considers the question of whether or not a P5-free graph can be 4-colored in polynomial time. It is known that a connected P5-free graph G must have either a dominating clique or a dominating P3. Thus, when considering the 4-coloring question, we have three cases of interest: either G has a dominating K4, a dominating K3, or a dominating P3. In this paper, we demonstrate a polynomial time approach for determining whether or not a P5-free graph G with a dominating K4 can be 4-colored.
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