Vincent Vatter
Professor University of Florida
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Areas of Expertise
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Articles
Three coloring via triangle counting
ArXivZachary Hamaker and Vincent Vatter
2022-03-15
In the first partial result toward the Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths four through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces.
Counting parity palindrome compositions
ArXivVincent Vatter
2021-09-27
A composition of the positive integer n is an ordered sequence of positive integers (its parts) that sum to n. A parity palindrome composition (ppc) is a composition that reads the same both forward and backward when reduced modulo 2; for example, 32141 is a ppc of 11. For brevity, we do not write a + between the parts of our compositions. Andrews and Simay [1] showed that ppcs have a surprisingly simple formula; see also Just [2]. We give a recursive proof.
Social Distancing, Primes and Perrin Numbers
Math HorizonsVincent Vatter
2021-08-20
This observation allows us to derive a recurrence for an. Given a full table with n chairs and at least two customers, we can add either one or two empty chairs and an occupied chair after the last customer (the customer sitting in the occupied seat with the greatest number) to obtain a full table with either n+ 2 or n+ 3 chairs.
Bijective proofs of proper coloring theorems
The American Mathematical MonthlyBruce Sagan and Vincent Vatter
2021-05-22
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits and in terms of acyclic orientations with compatible colorings.
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