不包含｛4，5，7｝－圈平面图是3－可染的

摘要

This article according to the several properties of minimal counterexample that planar graph without 4 and 5 7-cycles is 3-colorable .Then according to theorem that planar graph without 4 and 5 6 7-cycles is 3-colorable ,Finally using discharging rule that Planar graphs without cycles of length 4 ,5 ,7 are 3-colorable .%文章根据不包含｛4，5，7｝－圈平面图是3－可染极小反例的几个性质。接着根据每个不包含｛4，5，6，7｝－圈的平面图是3－可染的定理，利用discharging规则证明每个不包含｛4，5，7｝－圈平面图是3－可染的。