Improper 3-Coloring And Improper 3-Choosable Of Plane Graphs

In 1976,Steinberg proposed a famous conjecture:planar graphs without 4-cycles and 5-cycles are 3-colorable.This conjecture subsequently attracted widespread attentions,and a counterexample to the conjecture was given by Cohen-Addad until 2017.In the process,the scholars gave the weakened form of the Steinberg conjecture:planar graphs without 4-cycles and 5-cycles are(1,0,0)-colorable.But this problem has not been com-pletely solved so far.In 2009,Xu and Zhang's conjecture:planar graphs without adjacent 3-cycles are(3,1)*-choosable.This conjecture is still open so farWe study some related problems.The article is divided into three chapters.In the first chapter,we will introduce some basic concepts and status of improper 3-coloring and improper 3-choosable of planar graphs.In Chapter 2,we will prove that planar graphs without 4-,5-and 8-cycles are(1,0,0)-colorable.In Chapter 3,we will prove that planar graphs without adjacent k-cycles are(3,1)*-choosable,where k ? {3,4,5}.