Every Planar Graph Without Adjacent Triangles And 6-cycle Is (2,2,0)-colorable

Let c-₁,c₂,…,c_k be k nonnegative integers, and G be a simple planar graph. A （c₁, C₂,…, c_k）-coloring of G is a mapping c:V（G）â†'{1,2,…, k} such that for every 1≤i≤k, G[V_i] has maximum degree at most ci, where G[V_i] denotes the subgraph induced by the vertices colored i. Borodin and Raspaud conjecture that every planar graph without intersecting triangles and 5-cycle is 3-colorable. we prove that every planar graph without adjacent triangles and 6-cycle is （2,2,0）-colorable. The concrete content is in the following:· In chapter 1, we introduce the background significance of the research, and the problem we will solve. Based on this research background and profound dis-cussion, by using deep-going analysis, it fully shows the main work’s necessity and innovation.· In chapter 2, we give some necessary definition and lemmas.· In chapter 3, we introduce some necessary definition and lemmas of the house in G.· In chapter 4, we give the discharging rules and some relevant lemmas.· In chapter 5, we give the proof of the nonnegative of the final charge of vertex and faces in G.· In chapter 6, we summary the paper and give prospects in the future.