A Thought On Barnette's Conjecture Via Cochain Complex

In 1969,D.W.Barnette proposed a famous conjecture that every cubic 3-connected planar bipartite graph(called Barnette graph)is Hamiltonian.In this article,we look at the history of Barnette's Conjecture and discuss it via cochain complex.By Steinitz's theorem,we know Barnette graph can correspond to a 1-skeleton of a 3-dimensional simple convex polytope.Then any 3-dimensional simple convex polytope can correspond to a simplicial2-dimensional sphere(called dual simplicial sphere).By Kelmans'conclusion,we know Barnette's Conjecture can be equivalent to the description on flag even simplicial 2-dimensional sphere,so we can consider Stanley-Reisner ring on it,and then construct the cohomology chain with 2-modules coefficient structure(called the cohomology chain of vertices subset).Based on some results of the cohomology groups,we give another equivalent description on Barnette's Conjecture:For any flag simplicial 2-dimensional sphere whose vertices are all of even degrees,its vertices can be divided into two parts(1 and(1⁽(8) such that cohomology chain corresponding to each vertices subset is exact in 1 and 2 dimensions,and the dimensions of the first three cochain groups satisfies arithmetic sequence relationship.The first chapter is the introduction,mainly introduces the research background of the article;the second chapter is the preparation knowledge required for the article;the third chapter introduces the previous achievements and the current progress of Barnette's Conjecture;chapter four gives some equivalent descriptions of Barnette's Conjecture via algebraic topology;chapter five describes the significance of the research and the problems to be solved in the future.