**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,