Signed Planar Graphs Without 4-,5-,7-and 8-Cycles Are 3-Colorable

All graphs in this paper are finite and simple.Let G =(V(G),E(G))is a graph and ?:E(G)? {1,-1} is a mapping,then the pair(G,?)is called a signed graph,and ? is called a signature of G.Let e is an edge in G,e is positive if?(e)?1;e is negative if ?(e)=-1.Signed graph(G,?)in k colors to be a mapping V(G)? {±1,±2,...,±k/2 if k is even,or V(G)? {0,±1,...,±k-1/2} if k is odd,such that for every edge e = uv ? E(G),c(u)? ?(uv)c(v).A signed graph(G,?)is k-colorable if it exists a k-coloring.Study of graph coloring begins with the famous "Four Color Conjecture",but the theorem has not been proved by elegant math-analysis so far.Therefore,many experts began to study the 3-coloring problem of the floor plan in order to find a mathematical proof method.The 3-coloring problem of the plan was first introduced by planar graph in 1959,Grotzsch proved that every triangle-free planar graphs is 3-colorable?Actually,his condusion also holds true for signed graph.In 1976 Steinberg came up with a conjecture:each planar graphs without 4,5-cycles are 3-colorable,but this conjecture was proved to be false at Cohen-Addad,Hebdige,Kral,Li,and Salgado in 2017.Steinberg's conjecture is also false for signed graph,but the problem also exists.At present,there are a lot of problems on the coloring of signed graph.In this paper,we prove that signed planar graphs without 4-,5-,7-and 8-Cycles are 3-Colorable.· In chapter 1,we introduce the background of the research,the latest progress,and the problem we have solved in this paper.· In chapter 2,we introduce the symbols and basic concepts involved in this paper.· In chapter 3,we give some reducible configurations of G and proof.· In chapter 4,we give the discharging rules and prove that the final charge of vertices and faces are positive in G.