Minimum Coprime Labelings of Generalized Petersen and Prism Graphs

Threshold Colorings of Prisms and the Petersen Graph

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 th…

Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $Π_1,\ldots,Π_k$, where $Π_i$, $i\in [k]$, is an $i$-packing. The following conjecture is posed and studied: if $G$ is a subcubic graph, then $χ_ρ(S(G))\le 5$, where $S(G)$ is the subdivision of $G$. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if $G$ is a generalized prism of a cycle, then $G$ is $(1,1,2,2)$-colorable if and only if $G$ is not the Petersen graph. The validity of the conjecture is fu…

各种图的总素数标记

Total Prime Labelings of Various Graphs

本文提出了一种总素数标记方法，用于各种图类，证明了蛇形图、书形图、棱柱图等是总素数图。