Reducible Configurations and So On: The Final Years of the Four Color Theorem

The Four Color Theorem is in a set of mathematical questions that are very simple to state but amazingly complex to answer. It goes as follows, "given any map, are any more than 4 colors required to color the map in such a way that no two areas which share a border also share a color?"(2). It was thought to be proven by Alfred Kempe for nearly a decade using a unique but unsuccessful process later referred to as Kempe chains. It wasn't until 1913, with George Birkhoff's treatment of reducibility, was true progress from the "proof" of Kempe to be made. From here, Heinrich Heesch explored reducibility with an improvement on the established A, B, and C-reducibilities, finding something algorithmically sound in D-reducibility and his subsequent discharging methods. Then Karl Durre introduced the first, somewhat rudimentary, computer program of D-reducibility. From here the extensive use of the super computers of the era helped seal the fate of the long unfinished theorem, with Wolfgang Haken and Kenneth Appel at the helm. We seek to examine the history of this theorem from the proof of Kempe to the utilization of reducible configurations and discharging methods of Durre and Heesch and into the eventual proof of the theorem itself

Solitaire Clobber

Clobber is a new two-player board game. In this paper, we introduce the one-player variant Solitaire Clobber where the goal is to remove as many stones as possible from the board by alternating white and black moves. We show that a checkerboard configuration on a single row (or single column) can be reduced to about n/4 stones. For boards with at least two rows and two columns, we show that a checkerboard configuration can be reduced to a single stone if and only if the number of stones is not a multiple of three, and otherwise it can be reduced to two stones. We also show that in general it is NP-complete to decide whether an arbitrary Clobber configuration can be reduced to a single stone.Comment: 14 pages. v2 fixes small typ

Edge-coloring via fixable subgraphs

Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration; these configurations are called \emph{reducible} for that theorem. (A \emph{configuration} is a subgraph $H$, along with specified degrees $d_G(v)$ in the original graph $G$ for each vertex of $H$.) We give a general framework for showing that configurations are reducible for edge-coloring. A particular form of reducibility, called \emph{fixability}, can be considered without reference to a containing graph. This has two key benefits: (i) we can now formulate necessary conditions for fixability, and (ii) the problem of fixability is easy for a computer to solve. The necessary condition of \emph{superabundance} is sufficient for multistars and we conjecture that it is sufficient for trees as well, which would generalize the powerful technique of Tashkinov trees. Via computer, we can generate thousands of reducible configurations, but we have short proofs for only a small fraction of these. The computer can write \LaTeX\ code for its proofs, but they are only marginally enlightening and can run thousands of pages long. We give examples of how to use some of these reducible configurations to prove conjectures on edge-coloring for small maximum degree. Our aims in writing this paper are (i) to provide a common context for a variety of reducible configurations for edge-coloring and (ii) to spur development of methods for humans to understand what the computer already knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure

Coloring count cones of planar graphs

For a plane near-triangulation G with the outer face bounded by a cycle C, let n⋆G denote the function that to each 4-coloring ψ of C assigns the number of ways ψ extends to a 4-coloring of G. The block-count reducibility argument (which has been developed in connection with attempted proofs of the Four Color Theorem) is equivalent to the statement that the function n⋆G belongs to a certain cone in the space of all functions from 4-colorings of C to real numbers. We investigate the properties of this cone for |C|=5, formulate a conjecture strengthening the Four Color Theorem, and present evidence supporting this conjecture