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A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings. (arXiv:1810.07302v1 [math.GT])
来源于:arXiv
In this paper, we prove a new cohomology theory that is an invariant of a
planar trivalent graph with a given perfect matching. This bigraded cohomology
theory appears to be very powerful: the graded Euler characteristic of the
cohomology is a one variable polynomial (called the 2-factor polynomial) that,
if nonzero when evaluated at one, implies that the perfect matching is even.
This polynomial can be used to construct a polynomial invariant of the graph
called the even matching polynomial. We conjecture that the even matching
polynomial is positive when evaluated at one for all bridgeless planar
trivalent graphs. This conjecture, if true, implies the existence of an even
perfect matching for the graph, and thus the trivalent planar graph is
3-edge-colorable. This is equivalent to the four color theorem---a famous
conjecture in mathematics that was proven using a computer program in 1970s.
While these polynomial invariants may not have enough strength as invariants to
prove such a co 查看全文>>