Total Coloring Of Planar Graph Without4-Fans

Let G=(V, E, F) be a graph with the set of vertices V, the set of edges E and the set of faces F. For a plane graph G, we use Δ and δ to denote its maximum degree and minimum degree. A total k-coloring of G is a mapping φ:V∪Eâ†'{1,…, k} such that φ(u)≠m(u) whenever u and v are two adjacent or incident elements of V U E. The total chromatic number of G is the minimum positive integer k so that G has a k-totally-coloring. G is totally k-colorable if it admits a total k-coloring. Clearly, the total chromatic number of G is at least Δ+1.In1960s, Vizing and Behzad independently conjectured that every simple graph is totally (Δ+2)-colorable. This is the famous total coloring conjecture, in short, TCC. So far, the only case has not yet been solved for TCC is Δ=6. For plane graph with Δ=6, Wu J L and Wang Y Q etc. independently showed TCC is true when the graph has no ki-cycle for ki∈{3,4,5,6}. Recently, Roussel showed that if every vertex of a plane graph with maximum degree6is missing from some kv-cycle for kv∈{3,4,5,6,7,8}, then the graph is totally8-colorable.For planar graph, many graphs are totally (Δ+1)-colorable. Firstly, Borodin showed that plane graph with Δ≥11is (Δ+1)-colorable; then Wang W F improve it to Δ=10; after that Kowalik improve it to Δ=9. As for plane graph with Δ≤8, it seems hard to obtain the simple result as above. Consider the totally9-colorable of plane graph with maximum degree8, Hou J F prove that plane graph with Δ≥8without4-cycle is totally9-colorable. Then the additional condition without4-cycle is generally reduced to without5-cycle; without6-cycle; without intersecting1-fans; without2-fans and without3-fans etc..Based on the previous work, this thesis studies the TCC and problems of planar graphs. By exploring new reducibility of minimal counterexample, we use discharging method to prove that:(1) Plane graphs with maximum degree6and without4-fans are totally8-colorable.(2) Plane graphs with maximum degree8and without4-fans are totally9-colorable.where a4-fan in a plane graph consists of4consecutive3-faces intersecting at a vertex, similarly, k consecutive3-faces intersecting at a vertex is called a k-fan. This improves some known results on topic of related problems.