New Proof Advances Four-Color Theorem and Graph Coloring
Mathematicians have produced a new computer-assisted proof of the four-color theorem, published online in March 2026 and highlighted on September 10. The result, by a team including Mikkel Thorup, Carsten Thomassen, Ken-ichi Kawarabayashi and others, reconfirms that any planar map can be colored with four colors so no adjacent regions share a color. It stands out for delivering a far more efficient algorithm that colors planar graphs in near-linear O(n log n) time rather than the previous quadratic O(n²) bound.
The four-color theorem was first proved in 1976 by Appel and Haken using computers to check thousands of configurations, a method initially controversial. A streamlined 1997 proof reduced the configurations to 633. The new work identifies an unavoidable set of 8,202 configurations concentrated in “flat” regions of planar graphs. These can be reduced in parallel without mutual interference, yielding both a fresh proof and structural insights into planar graphs.
Key uncertainties remain around the complexity of the computer verification itself and whether a fully human-readable, non-computer proof will ever emerge. The parallel-reduction technique may transfer to other graph-theory problems, but its broader applicability is still untested. The result strengthens algorithmic tools while leaving open deeper questions about why four colors always suffice.
Sources: Quanta Magazine.
Nature-Inspired Foam Delivers Pigment-Free Whiteness and Water Repellency
Researchers at Kyoto University and collaborators announced on September 10 a foam material that achieves intense structural whiteness and strong water repellency without titanium dioxide pigments or PFAS chemicals. The process, called Deep Foam Photolithography, uses light to fragment a polymer followed by mild solvent treatment, creating a porous network that scatters light and a rough surface that sheds water.
Structural color and whiteness occur naturally in snow, clouds, sea foam, and certain plant tissues or insect scales through microscopic architecture rather than pigments. The team applied the method to commercial polymers and fabrics, achieving 20,000 DPI resolution printable materials. This addresses regulatory pressure on TiO₂ (banned as a food additive in the EU) and persistent fluorinated compounds.
Tensions include scaling the process to industrial volumes and costs competitive with conventional coatings, plus long-term durability under real-world wear and weathering. While the approach embeds function in physical structure rather than chemistry, performance consistency across different base polymers and environmental conditions remains to be fully quantified. The work opens routes to lighter, more sustainable packaging, textiles, and coatings.
Sources: ScienceDaily, Kyoto University iCeMS.
Mercury Has Shrunk More Than Previously Estimated
A study published September 10 in Geophysical Research Letters finds that Mercury’s radius has contracted 10–30 percent more than earlier estimates, equivalent to an additional several kilometers of shrinkage since formation. Impact-crater debris had obscured many of the planet’s characteristic wrinkle ridges and scarps that record cooling-driven contraction.
Mercury’s large iron core relative to its size drives substantial cooling and shrinkage over 4.5 billion years. Prior analyses based on MESSENGER data underestimated the total because rough, debris-covered terrain hid structures. Accounting for this bias raises the radial contraction to roughly 7–12 km, bringing observations into better agreement with thermal-evolution models.
Uncertainties center on the exact contribution of hidden small-scale features and the precise core composition (sulfur or silicon content) that governs cooling rate. Higher-resolution data expected from the BepiColombo mission later in 2026 should refine the numbers. The revised figure tightens constraints on rocky-planet interior evolution but does not yet resolve remaining discrepancies between models and surface geology.
Sources: Phys.org, Geophysical Research Letters coverage.