Threshold Colorings of Prisms and the Petersen Graph

TL;DR

This paper investigates a specific graph coloring problem called threshold coloring, proving that prisms and the Petersen graph are total threshold colorable, while Moebius ladders are not, highlighting the complexity of characterizing such graphs.

Contribution

It establishes that prisms and the Petersen graph are total threshold colorable, and shows Moebius ladders are not, revealing the non-finite nature of characterizing total threshold colorability.

Findings

Prisms are total threshold colorable.

The Petersen graph is total threshold colorable.

Moebius ladders are not total threshold colorable.

Abstract

Let $G$ be a graph, $r \geq t$ integers, and $N \subseteq E(G)$. An $(r,t)$-threshold-coloring of $G$ with respect to $N$ is a mapping $c: V(G) \rightarrow \{0,\ldots,r-1\}$ such that $|c(u)-c(v)| \leq t$ for every $uv \in N$ and $|c(u)-c(v)|>t$ for every $uv \in E(G) \setminus N$. A graph is total threshold colorable if there exist integers $r,t$ such that for every $N \subseteq E(G)$, $G$ admits an $(r,t)$-threshold-coloring with respect to $N$. We show that every prism is total threshold colorable, and that the Petersen graph is total threshold colorable. In contrast to this fact we show that Moebius ladders are not total threshold colorable, from which it follows that there is no characterization of being total threshold colorable in terms of a finite set of forbidden subgraphs.