A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings

Abstract

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 Tait polynomial. We conjecture that the Tait 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 with the aid of a computer in the 1970s. While these polynomial invariants may not have enough strength as invariants to prove such a conjecture directly, it is hoped that the strictly stronger cohomology theory developed in this paper will shed light on these types of problems.

The 2-Factor Polynomial Detects Even Perfect Matchings

“…For the 2-factor polynomial, first note that if G has a bridge, then it must be a perfect matching edge (cf. [2]). Resolving this perfect matching edge shows that the 2-factor polynomial has a (z − 1) factor, and therefore G : M 2 (1) = 0 (again, see [2]).…”

“…[2]). Resolving this perfect matching edge shows that the 2-factor polynomial has a (z − 1) factor, and therefore G : M 2 (1) = 0 (again, see [2]). We form another graph T by placing a vertex in each face of D and joining two vertices when their associated faces are separated by an edge of M .…”

“…In [2], the first author developed a cohomology theory of a planar trivalent graph equipped with a perfect matching, and observed that the graded Euler characteristic of that cohomology is a polynomial invariant called the 2-factor polynomial. Given a planar trivalent graph G with a perfect matching M , a perfect matching drawing Γ for the pair (G, M ) is a topological embedding of the graph in the 2-sphere that marks the perfect matching edges.…”

“…In Section 3 of [2], it was shown that the 2-factor polynomial is independent of the choice of perfect matching drawing, and thus is an invariant of planar trivalent…”

“…The calculation of these examples also serve as a way to introduce the cube of resolutions, which we will use throughout this paper (cf. [2]). Briefly, to construct a cube of resolutions for (P 3 , L), first resolve each perfect matching of the graph into one of the two resolutions shown in the first equation of Figure 1 to get a state.…”