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Below is my draft paper headlines (planning to present at ICM 2014) The Big Bang Theory of Planar Graph Coloring: Solution of the Three Color Problem Soon after Big Bang blast: 1. All graphs are triangulated and have chromatic number 4 2. Some triangulated planar disk graphs are “even” triangulated and hence with chromatic number 3. 3. Triangulated planar graphs with holes. If all holes satisfy parity symmetric conditions then the graph is 3 colorable otherwise 4 colorable by 4CT. Here PS property is violated only there is a “crash” of triangulated chain with an edge. 4. Planar graphs formed by quasi uniquely (x,x)- and (x,y) gadgets and chromatic number is 3 only if there is no crash between any two gadgets. 5. Planar graph with weakly connected triangles e.g., two triangles with common vertex. Such a planar graph has chromatic number 4 for all three colorings iff there is an non-triangle edge or vertex crashed with Kempe-tangling. 6. Planar graphs with disjoint triangles such that any two triangle apart each other by the distance at most 4. Such planar graphs have chromatic number 4 iff there is crash with quasi edges of an induce K_4. Otherwise the chromatic number is 3. 7. Planar graphs without triangles are all three colorable. 8. Planar bipartite graphs have chromatic number 2. 9. Trees are two colorable. 10. Graphs with no edges are mono-chromatic (all disjoint vertices colored by RED!) 11. End of the time (nothing to color) in the Big Bang Theory of Planar Graph Coloring. Theorem: A planar graph is 3 colorable iff there is no “crash” of two induce quasi uniquely three colorable subgraphs.
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