Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs

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 [13] which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decompositions using graph labelings. 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 b̄ ∈ B such that {b̄, 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(b̄) = 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 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.


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 [13] which states that if T is any tree with m edges then the complete graph K 2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decompositions using graph labelings. 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, Further, Blinco et al. [6] proved a significant result that the complete graph K 2cn+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 . 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 K m,n , where m ≥ 3, n ≥ 4 plus an edge admits γ-labeling. ii) For t ≥ 2, t-levels shadow graph of the path P dn+1 admits d-divisible α-labeling for any admissible d and n ≥ 1.
Further, we discuss related open problems.

Introduction
Terms which are not defined here can be found in [15]. In an attempt to settle the Ringel's conjecture [13] which states that if T is any tree with m edges then the complete graph K 2m+1 can be decomposed into 2m + 1 copies of T , in his classical paper [14], Rosa introduced a series of labelings called α, β, σ, ρ-labeling. Let G be a graph with n edges. A one-to-one function g from V (G) to {0, 1, 2, . . . , n} is called a β-labeling of G if {|g(u) − g(v)|/{u, v} ∈ E(G)} = {1, 2, . . . , n}. A β-labeling g of a graph G with n edges is called an α-labeling if there exists an integer k such that for every edge {u, v} ∈ E(G) either g(u) ≤ k < g(v) or g(v) ≤ k < g(u). Given two vertices u and v by uv we denote the edge {u, v}.
It is clear that α-labeling is a stronger version of β-labeling. β-labeling was later called as graceful labeling by Golomb [12] and this term is most widely used now. ρ-labeling is weaker version of graceful labeling. The precise definition of ρ-labeling is given below. Let G be a graph with n edges. A one-to-one function g from V (G) to {0, 1, 2, . . . , 2n} is called a Further, Rosa [14] proved the following two significant theorems.
Theorem 1.1. Let G be a graph with n edges. Then there exists a cyclic G-decomposition of the complete graph K 2n+1 if and only if G has a ρ-labeling.
If G is a graph with n edges that has an α-labeling, then the complete graph K 2cn+1 can be cyclically decomposed into subgraphs isomorphic to G, where c is an arbitrary natural number.
The interesting part of α-labeled graphs with n edges is that they not only decompose complete graphs K 2cn+1 but also decompose the complete bipartite graphs K an,bn . This interesting result proved by El-Zanati and Vanden Eynden [9] is precisely stated in the following theorem. Theorem 1.3. If a graph G with n edges has an α-labeling then there exists a cyclic decomposition of the complete bipartite graph K an,bn into subgraphs isomorphic to G, where a and b are arbitrary positive integers.
These results attracted many researchers to significantly contribute in theory of graph decompositions using graph labelings. It is clear from the definition of α-labeling that if a graph G admits α-labeling then it must be necessarily bipartite. This restriction prompted Blinco et al. [6] to introduce γ-labeling in order to achieve cyclic G-decompositions in K 2cn+1 , where G is a non-bipartite graph, c is any positive integer and n is the number of edges of the graph G. 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} andb ∈ B such that {b, c} is the unique edge joining an element of B to c, Motivated by the above result of Blinco et al. [6], the almost-bipartite graphs P n + e, n ≥ 4, K m,n +e, m ≥ 2, n > 2, C 2k+1 , k ≥ 2, C 2m +e, m > 2, C 3 ∪C 4m , m > 1, C 2k+1 ∪C 4n+2 , k ≥ 1, n ≥ 1 are found to have γ-labeling (refer [5], [6], [7], [8], [10]). (A graph is said to be almost-bipartite if the removal of a particular edge makes the graph bipartite). For survey on γ-labeling refer the survey on graph labelings by Gallian [11]. Motivated by the results of Blinco et al. [6], in this paper we prove that for t ≥ 2, disjoint union of t copies of the complete bipartite graph K m,n , where m ≥ 3, n ≥ 4 plus an edge admits γ-labeling.
Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called d-divisible graceful labeling as a tool to obtain cyclic G-decomposition in complete multipartite graphs. Let G be a graph of size e = d . m. An injective function (m + 1)} is called as a d-divisible graceful labeling of the graph G. A d-divisible graceful labeling of a graph G can exist only if d is a divisor of the size e of G, hence, for this reason, any divisor d of e is said to be admissible for the existence of a d-divisible graceful labeling of G. 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.
can be cyclically decomposed into copies of the d-divisible graceful labeled graph G, where e is the size of the graph G. Theorem 1.6. (Anita Pasotti [4]) The complete multipartite graph K ( e d +1)×2dc can be cyclically decomposed into copies of the d-divisible α-labeled graph G, where e is the size of the graph G and c is any positive integer.
In the literature survey [11], one can observe that a very few families of graphs are identified to have d-divisible α-labeling. Anita Pasotti [4] has proved that path and star admit d-divisible α-labeling for any admissible d. She [3] also proved that for any integer k ≥ 1 and m ≥ 2, C 4k × P m admits (2m − 1)-divisible α-labeling. In [1] and [2], Anna Benini and Anita Pasotti proved the following results. A hairy cycle of size e admits an e-divisible α-labeling if and only if it is bipartite. The hairy cycle H(2t, λ) admits d-divisible α-labeling for any admissible d. The ladder L 2k has 2-divisible α-labeling if and only if k is even.
Inspired by the decomposition theorems proved by Anita Pasotti, in this paper we prove that for t ≥ 2, t-levels shadow graph of the path P dn+1 admits d-divisible α-labeling for any admissible d and n ≥ 1. t-levels shadow graph of a graph is defined as follows. t-levels shadow graph of a graph G, denoted S t (G) is obtained by taking t ≥ 2 copies G 1 , G 2 , . . . , G t of G and joining each vertex v ij in G i to the copies of its adjacent vertices in G i+1 , for 1 ≤ j ≤ n and 1 ≤ i ≤ t − 1, where n = |V (G)|.
2 γ-labeling of disjoint union of complete bipartite graphs plus an edge In this section we prove that disjoint union of t copies of the complete bipartite graph K m,n , where m ≥ 3 and n ≥ 4 plus an edge admits γ-labeling.
Theorem 2.1. For t ≥ 2, disjoint union of t copies of a complete bipartite graph with one part containing at least three vertices and another part containing at least four vertices, plus an edge admits γ-labeling.
Proof. Consider the complete bipartite graph K m,n , where m ≥ 3, n ≥ 4.
. . , v in } be the two parts of the i-th copy K i m,n of the complete bipartite graph K m,n .
Clearly, U and V are the two parts of the disjoint union of the t copies of K m,n , denoted by Join the vertices v 11 and v 12 by an edgeê.
Denote the new graph thus obtained by ( First we define the labels of the vertices in the set U in the following way. For 1 ≤ j ≤ m, define g(u 1j ) = 2(j − 1) and g(u 2j ) = 2j + 1.
Now we define the labels of the vertices in the set V in the following manner.
We define the labels of the vertices v 2k , for k, 2 ≤ k ≤ n in two cases depending on n is even or odd.
Case 1. n is even define the labels of the vertices v ik , for each k, 2 ≤ k ≤ n in the following way.
We prove that the vertex labels of the graph ( t i=1 K i m,n ) +ê are distinct depending on n is even or odd.
Hence the vertex labels of the graph ( t i=1 K i m,n ) +ê are distinct.
Observation 2. Edge labels of ( The edge v 11 v 12 has the label N .
We prove that the edge labels of t i=1 K i m,n are distinct in two cases depending on n is even or odd.
Case i. n is even The labels of the edges in the first copy K 1 m,n can be arranged as a sequence, S 11 : ((N − 1, N − 2, N − 3, . . . , N + 2m + 1 − mn, N + 2m − mn), (2m, 2m − 1, . . . , 2, 1)). For each i, 2 ≤ i ≤ t, the labels of the edges in the i th copy K i m,n can be arranged as a sequence, The labels of the edges in the above sequences together with the label of the edge v 11 v 12 , |g(v 11 ) − g(v 12 )| = N can be rearranged as a monotonic decreasing sequence S : (N, N − 1, N − 2, . . . , 3, 2, 1). Thus the edge labels are distinct when n is even.
Hence the edge labels of the graph ( Observation 3. g is a γ-labeling.
In order to prove that g is a γ-labeling, we partition the vertex set V ((

d-divisible α-labeling of t-levels shadow graph of path
In this section we prove that for t ≥ 2, t-levels shadow graph of the path P dn+1 , S t (P dn+1 ) with d ≥ 1, n ≥ 1 admits d-divisible α-labeling for all d ≥ 1.
Theorem 3.1. For t ≥ 2, the t-levels shadow graph of the path P dn+1 , S t (P dn+1 ) with d ≥ 1 and n ≥ 1 admits d-divisible α-labeling for all d ≥ 1.
For the convenience, we let Then the t-levels shadow graph of the path P dn+1 , S t (P dn+1 ) has the vertex set Therefore, |V (S t (P dn+1 ))| = t|V (P dn+1 )| = t(dn + 1). By the definition of the t-levels shadow graph of the path P dn+1 , the graph S t (P dn+1 ) can be visualised as t copies of the path P dn+1 and a pair of t − 1 copies of P dn+1 which connect the vertices of the copies G i and G i+1 of the path P dn+1 , 1 ≤ i ≤ t − 1. Therefore, |E(S t (P dn+1 ))| = tdn + 2(t − 1)dn = (3t − 2)dn.
For all the remaining vertices of S t (P dn+1 ) we define g depending on d = 1 and d > 1.
When d = 1 define g as follows.
where s = dn, if dn + 1 is even dn + 1, if dn + 1 is odd, then the above sequence forms a strictly increasing sequence. Hence the vertex labels of S t (P dn+1 ) are distinct. From the above arrangement of vertex labels observe that dn+1) ), when dn + 1 is even; while when dn + 1 is odd, We prove that the edge labels of S t (P dn+1 ) are distinct depending on d = 1 and d > 1. Case 1. d = 1 When n is even, the edges of the graph S t (P dn+1 ) can be arranged as the following sequence, When n is odd, the edges of S t (P dn+1 ) can be arranged as the following sequence, Then from the definition of g for both the cases we have the corresponding edge label sequence, (N, N − 1, N − 2, . . . , 3, 2, 1). Hence, it is clear that the edge labels are distinct. Therefore, when d = 1, g is a 1-divisible α-labeling of S t (P dn+1 ). That is, g is an α-labeling of the graph S t (P dn+1 ). Case 2. d > 1 In order to show that the edge labels of the edges of S t (P dn+1 ) are distinct, we partition the edge set of S t (P dn+1 ) into d subsets of the edge set of S t (P dn+1 ) and they are arranged as d sequences. Consequently, their corresponding edge labels are also arranged as d sequences.
When n is even then we consider the first edge sequence to be the following sequence When n is odd then we consider the first edge sequence to be the following sequence Then from the definition of g for both the cases we have the corresponding edge label sequence, When n is even then we consider the second edge sequence to be the following sequence When n is odd then we consider the second edge sequence to be the following sequence Then from the definition of g for both the cases we have the corresponding edge label sequence, When n is even then we consider the third edge sequence to be the following sequence When n is odd then we consider the third edge sequence to be the following sequence Then from the definition of g for both the cases we have the corresponding edge label sequence, In general, we consider the j th edge sequence, for 4 ≤ j ≤ d − 2 depending on n and j. Case i. n is even or n is odd and j is even Then we consider the j th edge sequence to be the following sequence Case ii. n and j are odd Then we consider the j th edge sequence to be the following sequence Then from the definition of g for all the above cases we have the corresponding edge label sequence, S j : ((d−j)(3t−2)n+d−(j +1), (d−j)(3t−2)n+d−(j +2), (d−j)(3t−2)n+d−(j +3), . . . , (d−(j +1))(3t−2)n+d−(j +2), (d−(j +1))(3t−2)n+d−(j −1), (d−(j +1))(3t−2)n+d−j). Now we consider the (d − 1) th edge sequence depending on n is even or odd. S d ). Then we observe that S forms a monotonically decreasing sequence. Also observe that none of the terms (d − 1)((3t − 2)n + 1), (d − 2)((3t − 2)n + 1), . . . , 3((3t − 2)n + 1), 2((3t − 2)n + 1), (3t − 2)n + 1 appear in the combined sequence S. Thus, g is a d-divisible α-labeling of S t (P dn+1 ) for any admissible d > 1. Therefore the graph S t (P dn+1 ) admits d-divisible α-labeling for any admissible d.

Discussion
In this section we pose two open problems for further research.
In Theorem 2.1 we have proved that for t ≥ 2, disjoint union of t copies of the complete bipartite graph K m,n plus an edge, ( t i=1 K i m,n ) +ê admits γ-labeling. In this direction investigating the following question will be useful for achieving a generalised result.
Is it true that disjoint union of t copies of an α-labeled graph G plus an edge, t ≥ 2, admits γ-labeling?
In Theorem 3.1 we have proved that for t ≥ 2, the t-levels shadow graph of the path P dn+1 with d ≥ 1, n ≥ 1 admits d divisible α-labeling for all d ≥ 1. It is evident that the path P dn+1 admits α-labeling for all d ≥ 1, n ≥ 1. This observation tempts us to ask the following question to understand d-divisible α-labeled graphs.
What are the α-labeled graphs whose t-levels shadow graph admits d divisible α-labeling for all values of d?