[Paper Review] Ladder and Subdivision of Ladder Graphs with Pendant Edges are Odd Graceful

M. I. Moussa, E. M. Badr|arXiv (Cornell University)|Apr 8, 2016

Graph Labeling and Dimension Problems7 citations

TL;DR

This paper introduces a new class of odd graceful labeling for ladder graphs and their variants, proving that ladder graphs with pendant edges (Ln + mk₁), their subdivisions (S(Ln) + mk₁), and subdivisions of triangular snakes with pendant edges (S(Δ-k snake) + mk₁) are all odd graceful. The authors establish explicit labeling schemes using arithmetic sequences and parity-based assignments to achieve bijective vertex and edge labelings satisfying the odd graceful condition.

ABSTRACT

The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder graph. In particular, we show that the ladder graph Ln with m-pendant Ln + mk1 is odd graceful. We also show that the subdivision of ladder graph Ln with m-pendant S(Ln) + mk1 is odd graceful. Finally, we prove that all the subdivision of triangular snakes (delt-k snake) with pendant edges S(delt-k snake)+ mk are odd graceful.

Motivation & Objective

• To extend the class of odd graceful graphs by introducing new constructions involving ladder graphs with pendant edges.
• To investigate the odd gracefulness of subdivided ladder graphs and triangular snake variants with pendant edges.
• To provide explicit labeling algorithms that satisfy the odd graceful labeling condition for the proposed graph families.
• To generalize existing results on graceful labeling to the odd graceful framework for structured graph families.

Proposed method

• Proposes a vertex labeling scheme using arithmetic sequences and parity-based assignments to ensure odd edge labels.
• Constructs the labeling for Ln + mk₁ by assigning consecutive odd integers to vertices in a way that edge labels (absolute differences) are distinct odd integers.
• Applies the same labeling strategy to subdivided ladder graphs S(Ln) + mk₁ by inserting vertices on edges and reassigning labels accordingly.
• Extends the method to triangular snakes (Δ-k snakes) with pendant edges by adapting the labeling to the snake's cyclic structure and added pendants.
• Verifies the odd graceful condition by ensuring all edge labels are distinct odd integers from 1 to 2m−1, where m is the number of edges.
• Uses inductive and constructive techniques to validate the labeling across all proposed graph families.

Experimental results

Research questions

• RQ1 Can ladder graphs with pendant edges be labeled as odd graceful?
• RQ2 Are subdivided ladder graphs with pendant edges odd graceful?
• RQ3 Can the odd gracefulness property be extended to subdivisions of triangular snakes with pendant edges?
• RQ4 What labeling strategy ensures distinct odd edge labels in these structured graph families?
• RQ5 How do vertex and edge label assignments maintain the odd graceful condition under graph modifications like subdivision and pendant attachment?

Key findings

• The ladder graph Ln with m-pendant edges (Ln + mk₁) is odd graceful for all m ≥ 1.
• The subdivision of the ladder graph S(Ln) with m-pendant edges (S(Ln) + mk₁) is odd graceful for all m ≥ 1.
• All subdivisions of triangular snakes (Δ-k snakes) with m-pendant edges (S(Δ-k snake) + mk₁) are odd graceful for all m ≥ 1.
• The authors provide explicit labeling constructions that achieve bijective odd edge labels without repetition.
• The labeling schemes are consistent across all tested graph families and satisfy the odd graceful condition by design.
• The results extend the known classes of odd graceful graphs to include structured variants of ladder and snake graphs with pendant edges.