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Full-Text Articles in Physical Sciences and Mathematics
The Four-Color Theorem And Chromatic Numbers Of Graphs, Sarah E. Cates
The Four-Color Theorem And Chromatic Numbers Of Graphs, Sarah E. Cates
Undergraduate Theses and Capstone Projects
We study graph colorings of the form made popular by the four-color theorem. Proved by Appel and Haken in 1976, the Four-Color Theorem states that all planar graphs can be vertex-colored with at most four colors. We consider an alternate way to prove the Four-Color Theorem, introduced by Hadwiger in 1943 and commonly know as Hadwiger’s Conjecture. In addition, we examine the chromatic number of graphs which are not planar. More specifically, we explore adding edges to a planar graph to create a non-planar graph which has the same chromatic number as the planar graph which we started from.
The Development Of An Effective Water-Soluble Receptor For Pyrene Derivative Dyes, Ashley Longstreet
The Development Of An Effective Water-Soluble Receptor For Pyrene Derivative Dyes, Ashley Longstreet
Undergraduate Theses and Capstone Projects
A receptor with enhanced water solubility was synthesized from an existing receptor to be used as a pyranine fluorescence quencher in endovesiculation detection assays. The existing receptor consisted of a cyclen core with four protruding arms containing an aryl nitro group at each end. Its poor water solubility limited the accuracy and precision of the endovesiculation assay because the low concentration of receptor did not match the concentration of pyranine. To increase the water solubility, ethoxy groups were attached to the ends of each arm by first selectively reducing the nitro group to an amine with NaBHU4, H2, and a …