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Equality of Parameters mu(D) and kappa(D) for Three Element Distance Sets

Undergraduate #253
Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics

Marcos Reyes - California State University, San Bernardino


Let $D$ be a set of positive integers. The distance graph $G (mathbb{Z},D)$ with distance set $D$ is the graph with vertex set $mathbb{Z}$, and two vertices $x$ and $y$ are adjacent if $|x-y|in D$. The chromatic number, fractional number, and circular chromatic number of distance graphs have been studied greatly over the past two decades. Closely related to these parameters is the density of a distance set $D$, $mu(D)$, and the parameter $kappa(D)$–where the latter is involved in the Lonely Runner Conjecture. It is known that $kappa(D)leq mu (D)$ for all distance sets $D$, and equality of these two parameters have been brought to light when $|D|leq 2$. Now, a natural question to ask is, “what about when $|D|=3$? Would equality still hold?” The evidence we have collected strongly favors the equality $mu(D)=kappa(D)$. Whether or not this is true for all 3-element distance sets remains an open question. I investigate the equality of the parameters with the set ${a,a^2,b}$ where $gcd(a,b)=1$ and $a$, $b$ are of opposite parities and also the set ${2, p, p+2}$ where $p$ and $p+2$ are twin primes. In conclusion, I explain some results for the parameter $kappa(D)$ as well.

Funder Acknowledgement(s): National Science Foundation grant: DMS-1247679

Faculty Advisor: Min-Lin Lo,

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This material is based upon work supported by the National Science Foundation (NSF) under Grant No. DUE-1930047. Any opinions, findings, interpretations, conclusions or recommendations expressed in this material are those of its authors and do not represent the views of the AAAS Board of Directors, the Council of AAAS, AAAS’ membership or the National Science Foundation.

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