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Applications of the Baudet–Schur–Van der Waerden

In: The New Mathematical Coloring Book

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  • Alexander Soifer

    (University of Colorado at Colorado Springs, College of Letters, Arts, and Sciences)

Abstract

At the end of Chap. 14 , I left you with the embedding in the plane of the 352,735-vertex Blanche Descartes graph by Paul O’Donnell. One may ask, would attaching longer k-cycles (k > 7) to the foundation vertices increase the graph’s girth while keeping the chromatic number at 4? The answer is no – not if k-cycles were attached to all k-element subsets of the foundation set – because some k-cycles would have two or more vertex intersection that could cut down the girth of the graph. We would get a chance to succeed at this construction if we were to dramatically limit the number of attached k-cycles, by, say, allowing at most a single point intersection for the k-subsets of the foundation, to which k-cycles are allowed to be attached. This is exactly what O’Donnell implemented.

Suggested Citation

  • Alexander Soifer, 2024. "Applications of the Baudet–Schur–Van der Waerden," Springer Books, in: The New Mathematical Coloring Book, edition 2, chapter 0, pages 643-646, Springer.
    • Handle: RePEc:spr:sprchp:978-1-0716-3597-1_48
    DOI: 10.1007/978-1-0716-3597-1_48
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