[Paper Review] Decompositions of complete multipartite graphs via generalized graceful labelings

Anna Miriam Benini, Anita Pasotti|arXiv (Cornell University)|Oct 16, 2012

Graph Labeling and Dimension Problems8 references3 citations

TL;DR

This paper introduces a novel construction of d-divisible α-labelings for caterpillars, hairy cycles, and cycles, leveraging generalized graceful labeling techniques to establish infinite families of cyclic decompositions in complete multipartite graphs. The key contribution is proving that such labelings imply the existence of cyclic Γ-decompositions of $K_{m imes n}$ for infinitely many pairs $(m,n)$, extending classical results on α-labelings and graceful labelings to broader graph families and parameters.

ABSTRACT

We prove the existence of infinite classes of cyclic G-decompositions of the complete multipartite graph, G being a caterpillar, a hairy cycle or a cycle. All the results are obtained by the construction of d-divisible $α$-labelings of G, introduced in [A. Pasotti, On d-graceful labelings, to appear on Ars Combin.] as a generalization of classical $α$-labelings, whose existence implies that one of graph decompositions.

Motivation & Objective

• To extend classical α-labeling and graceful labeling techniques to a generalized d-divisible framework for graph decompositions.
• To establish the existence of d-divisible α-labelings for specific graph families, including caterpillars, hairy cycles, and cycles.
• To prove that such labelings imply the existence of cyclic Γ-decompositions in complete multipartite graphs $K_{m imes n}$ for infinitely many parameter pairs $(m,n)$.

Proposed method

• Introduce and formalize the concept of d-divisible α-labeling as a generalization of classical α-labelings and odd graceful labelings.
• Construct explicit d-divisible α-labelings for caterpillars using a new labeling scheme called standard α_S-labeling.
• Use the existence of d-divisible α-labelings to derive cyclic Γ-decompositions of $K_{(rac{e}{d}+1) imes 2dn}$ via Theorem 1.4.
• Provide case-by-case labeling constructions for hairy cycles and cycles based on underlying caterpillar structures and edge types.
• Distinguish labeling rules based on edge type (pendant from A/B, cycle edges of form [x_s,y_{s-1}] or [x_s,y_s]) to ensure injectivity and correct difference sets.
• Explicitly construct 2-divisible α-labelings for hairy cycles $H(2t,1)$ when $t > 1$ is odd, splitting cases by $t \mod 4$.

Experimental results

Research questions

• RQ1 Can d-divisible α-labelings be systematically constructed for caterpillars for any admissible d?
• RQ2 Do hairy cycles and cycles admit d-divisible α-labelings for all admissible d values, particularly when all cycle vertices have the same degree?
• RQ3 What infinite families of complete multipartite graphs $K_{m imes n}$ admit cyclic decompositions into caterpillars, hairy cycles, or cycles?
• RQ4 How do d-divisible α-labelings generalize classical α-labelings and graceful labelings in the context of graph decompositions?
• RQ5 What structural properties of cycles and hairy cycles allow for the existence of d-divisible α-labelings across all admissible d?

Key findings

• A d-divisible α-labeling exists for every caterpillar and any admissible d, enabling the construction of infinite families of cyclic decompositions.
• The existence of a d-divisible α-labeling for a graph Γ of size e implies a cyclic Γ-decomposition of $K_{(rac{e}{d}+1) imes 2dn}$ for any positive integer n.
• Bipartite hairy cycles admit an odd α-labeling, and when all cycle vertices have equal degree, they admit a d-divisible α-labeling for any admissible d.
• For any positive integer k, the cycle $C_{4k}$ admits a d-divisible α-labeling for all admissible d values.
• The construction of 2-divisible α-labelings for $H(2t,1)$ when $t > 1$ is odd is explicitly provided, with separate cases for $t \equiv 1 \pmod{4}$ and $t \equiv 3 \pmod{4}$.
• When d=1, the constructed d-divisible α-labelings recover the classical α-labelings of Rosa for caterpillars and cycles.