Optimal radio labelings of the Cartesian product of the generalized Peterson graph and tree

TL;DR

The paper derives a sharp lower bound for the radio number of the Cartesian product $P_{m,k} \Box T$ of a generalized Petersen graph and a tree, tying the bound to the tree's total level and diameter via $rn(P_{m,k} \Box T) \ge (2mn-1)(d_t+\varepsilon) - 4mL(T)$. It provides necessary and sufficient characterizations for when the bound is tight, expressed through a vertex ordering with specific distance- and level-based constraints, and offers three practical sufficient conditions for tightness. The framework is then applied to the Petersen graph paired with a star, yielding an exact value $rn(P_{5,2} \Box K_{1,n}) = 10n+27$ for $n \ge 3$ by constructing explicit orderings. This advances understanding of radio labeling on Cartesian products and delivers concrete results for a classical graph family combination with trees.

Abstract

A radio labeling of a graph $G$ is a function $f : V(G) \rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest number $k$ such that $G$ has radio labeling $f$ with max$\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and tree. We present two necessary and sufficient conditions, and three other sufficient conditions to achieve the lower bound. Using these results, we determine the radio number for the Cartesian product of the Peterson graph and stars.

Figures (1)

• Figure 1: The Peterson graph $P_{5,2}$.

Theorems & Definitions (14)

• Definition 1.1
• Lemma 2.1
• Lemma 2.2
• Definition 2.4
• Definition 2.5
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3