Home > Journals > TAG > Vol. 7 > Iss. 2 (2020)

## Abstract

In his classical paper [14], Rosa introduced a hierarchical series of labelings called *ρ, σ, β* and *α* labeling as a tool to settle Ringel’s Conjecture which states that if *T* is any tree with *m* edges then the complete graph *K2m+1* can be decomposed into 2*m* + 1 copies of *T* . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced γ-labeling as a stronger version of *ρ*-labeling. A function *g* defined on the vertex set of a graph *G* with *n* edges is called a *γ*-labeling if

(i) *g* is a *ρ*-labeling of *G*,

(ii) *G* is a tripartite graph with vertex tripartition *(A, B, C)* with *C = {c}* and *¯b ∈ B* such that *{¯b, c}* is the unique edge joining an element of *B* to *c*

(iii) *g(a) < g(v)* for every edge *{a, v} ∈ E(G)* where a *∈ A*,

(iv) *g(c) - g(¯b) = n*

Further, Blinco et al. [6] proved a significant result that the complete graph *K2cn+1* can be cyclically decomposed into *c(2cn + 1)* copies of any γ-labeled graph with *n* edges, where *c* is any positive integer. Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called *d*-divisible graceful labeling as a tool to obtain cyclic *G*-decompositions in complete multipartite graphs. Let *G* be a graph of size *e = d . m.* A *d*-divisible graceful labeling of the graph *G* is an injective function ,em>g : V (G) → {0, 1, 2, . . . , d(m + 1) - 1} such that *{|g(u) - g(v)|/{u, v} ∈ E(G)} = {1, 2, . . . , d(m + 1) - 1}\{m + 1, 2(m + 1), . . . , (d - 1)(m + 1)}.* A *d*-divisible graceful labeling of a bipartite graph *G* is called as a *d*-divisible *α*-labeling of *G* if the maximum value of one of the two bipartite sets is less than the minimum value of the other one. Further, Anita Pasotti [4] proved a significant result that the complete multipartite graph *K (e/d +1)×2dc* can be cyclically decomposed into copies of *d*-divisible *α*-labeled graph *G*, where *e* is the size of the graph *G* and *c* is any positive integer *(K (e/d +1)×2dc* contains *e/d + 1* parts each of size *2dc)*. Motivated by the results of Blinco et al. [6] and Anita Pasotti [4], in this paper we prove the following results.

i) For *t ≥ 2,* disjoint union of *t* copies of the complete bipartite graph *Km,n*, where *m≥ 3, n ≥ 4* plus an edge admits *γ*-labeling.

ii) For *t ≥ 2*, *t*-levels shadow graph of the path *Pdn+1* admits *d*-divisible *α*-labeling for any admissible *d* and* n ≥ 1*.

Further, we discuss related open problems.

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

## Recommended Citation

Sethuraman, G. and Sujasree, M.
(2020)
"Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs,"
*Theory and Applications of Graphs*: Vol. 7:
Iss.
2, Article 2.

DOI: 10.20429/tag.2020.070202

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol7/iss2/2

*Supplemental Reference List with DOIs*