Cordial Labeling of Goldberg Snark and its related Graphs

TL;DR

This paper investigates cordial labeling for Goldberg Snark graphs, a Petersen-based family of snarks formed through vertex duplication, edge modification, and subdivision to yield graphs with $8n$ vertices and $12n$ edges. The authors prove that $G_n$ admits a cordial labeling when $n$ is odd and $n\neq 3$, and they provide explicit vertex-labeling patterns that balance vertex and edge label counts. They further show that path unions $G_n$ with $m$ copies, open stars $S(t,G_n)$, and one-point unions $P_n^{t}(t_n,G_n)$ are cordial under specified parity conditions, expanding cordial labeling to several Goldberg Snark-derived constructions. Collectively, the results establish cordiality for the Goldberg Snark family and its common graph-operations, contributing to understanding labeling properties in complex cubic graphs and snarks.

Abstract

In graph theory, a Snark is a connected, bridgeless, Cubic graph that cannot be edge-colored with only three colors. Additionally, to avoid some trivial cases, a Snark is typically required to have a girth of minimum five and a cyclic connectivity of minimum four. In this paper, we investigate the Cordial labeling, for one of the modified structures of Snark graph which is known as Goldberg Snark graph. Moreover, a few special forms of Goldberg Snark graph also admit the Cordial labeling.

Figures (14)

• Figure 1: Petersen Graph
• Figure 2: Goldberg Snark $G_{5}$
• Figure 3: Goldberg Snark $G_{7}$
• Figure 4: Subgraph $H_i$ of $G_{n}$
• Figure 5: Subgraph $H_j$ of $G_{n}$

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof