Radio gracefulness of Moore graphs and beyond

TL;DR

The paper investigates radio gracefulness for low-diameter graphs tied to Moore-graph theory, leveraging Hamiltonian-path structures in antipodal graphs to derive labeling schemes. It proves a new necessary and sufficient condition for radio gracefulness of bipartite graphs with diameter $3$, and computes the radio number for $(r,g)$-cages arising from generalized polygons, while establishing radio gracefulness for Erdős-Rényi polarity graphs and MMS graphs via constructive labeling techniques. The work shows that several cage families from generalized polygons are radio graceful in the diameter-6 case but not in diameter-8 or diameter-12 cases, with the antipodal graph structure playing a central role. These results clarify which highly symmetric networks admit efficient radio labelings and raise open questions about the completeness of Moore graphs as the only radio graceful cages.

Abstract

The study of radio graceful labelings is motivated by modeling efficient frequency assignment to radio towers, cellular towers, and satellite networks. For a simple, connected graph $G = (V(G), E(G))$, a radio labeling is a mapping $f: V(G) \rightarrow \mathbb{Z}^+$ satisfying (for any distinct vertices $u,v$) $$|f(u)-f(v)| + d(u,v) \geq diam(G)+1,$$ where $d(u,v)$ is the distance between $u$ and $v$ in $G$ and $diam(G)$ is the diameter of $G$. A graph is radio graceful if there is a radio labeling such that $f(V(G)) = \{1, \dots, |V(G)|\}$. In this paper, we determine the radio gracefulness of low-diameter graphs with connections to high-performance computing, including Moore graphs, bipartite Moore graphs, and approximate Moore graphs like $(r,g)-$cages, Erdős-Rényi polarity graphs, and McKay-Miller-Širáň graphs. We prove a new necessary and sufficient condition for radio graceful bipartite graphs with diameter $3$. We compute the radio number of $(r,g)-$cages arising from generalized $n-$gons. Additionally, we determine Erdős-Rényi polarity graphs and McKay-Miller-Širáň graphs are radio graceful.

Figures (3)

• Figure 1: Fano plane
• Figure 2: $G_I$ and $A(G_I)$
• Figure 3: Radio gracefulness of various $(r,g)-$cages

Theorems & Definitions (59)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: Saha and Basunia saha2020
• Theorem 2.4: Chartrand, Erwin, Harary, and Zhang chartrand2001radiolabeling
• Theorem 2.5: Dirac Dirac
• Theorem 2.6: Corollary of Moon and Moser Moon1963
• Definition 2.7
• Definition 2.8
• Remark 2.9
• Definition 2.10