arXiv subjects

Stephen Hartke

Publications and source records attributed to Stephen Hartke.

3 recordsLinked to original sources

arXiv2022

Relation between the correspondence chromatic number and the Alon--Tarsi number

We study the relation between the correspondence chromatic number and the Alon--Tarsi number, both upper bounds on the list chromatic number of a graph. There are many graphs with Alon--Tarsi number greater than the correspondence chromatic number. We present here a family of graphs with arbitrary Alon--Tarsi number, with correspondence chromatic number one larger. Keywords: correspondence coloring, Alon--Tarsi number AMS Mathematics Subject Classification: 05C15

math.CO↗

arXiv2012

On coloring of fractional powers of graphs

For $m, n\in \N$, the fractional power $\Gmn$ of a graph $G$ is the $m$th power of the $n$-subdivision of $G$, where the $n$-subdivision is obtained by replacing each edge in $G$ with a path of length $n$. It was conjectured by Iradmusa that if $G$ is a connected graph with $Δ(G)\ge 3$ and $1 ω(H^{3/5})$. However, we prove that the conjecture is true if $m$ is even. We also study the case when $m$ is odd, obtaining a general upper bound $χ(\Gmn)\leq ω(\Gmn)+2$ for graphs with $Δ(G)\geq 4$.

math.CO↗

arXiv2002

A general notion of visiblity graphs

We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results giving necessary properties of visibility graphs, and giving some examples of classes of graphs that can be so represented. Finally, we give some applications of the concept, and we provide potential avenues for future research in the area.

math.CO↗