A structural approach to the graceful coloring of a subclass of trees

Let M={1,2,..m} and G be a simple graph. A graceful m-coloring of G is a proper vertex coloring of G using the colors in M which leads to a proper edge coloring using M∖{m} colors such that the associated color of each edge is the absolute difference between their end vertices. The graceful chromatic number χg(G)= min {m:G admits a gracefulm− coloring }. We prove that 5≤χg(T)≤7, where T is a tree with Δ=4. Furthermore, we categorize the trees into three types along with its characterization and the related coloring algorithm are presented in this study.


Introduction
In 1967, Alexander Rosa [8] defined graph labeling as follows: A label from {0, 1, 2, ...} is given to every vertex or edge (or both) of a graph  = ( , ) satisfying certain conditions.Refer [5] for more details on graph labeling.Graceful labeling, a variant of graph labeling, was initially referred as -labeling by Alexander Rosa [8].Let  = {0, 1, ..., |()|}.A one-to-one function  is a graceful labeling from the vertex set of  to  resulting in another one-to-one function  * from the edges of  to  ⧵ {0}, where the label for every edge is the absolute difference between its end vertices.
The development in the field of graph theory led to the concept of graph coloring in which the vertex coloring and the edge coloring are the major topics of interest.A proper vertex coloring is assigning colors to the vertices of a graph  in such a way that different colors are assigned for adjacent vertices.In a similar manner, proper edge coloring is defined [2].Chromatic number (()) is the least number of colors required for proper coloring the vertices of a graph, where as the chromatic index ( ′ ()) is the least number of colors needed for proper coloring the edges of a graph.Graph coloring has extensive applications in various fields ( [2], [6], [9]).
Graceful coloring is an extension of graceful labeling [3] in which both vertex and edge colorings are involved.For any two positive integers ,  with  < , we define [, ] = {,  + 1, ...,  − 1, }.A graceful -coloring of  is a proper vertex coloring  of  using the colors in  = [1, 𝑚], where  ≥ 2, which leads to a proper edge coloring  * using  ⧵ {} colors such that the associated color of each edge is the absolute difference between their end vertices.The graceful chromatic number   () = min { ∶  admits graceful − coloring }.
For a subgraph  of ,   ( ) ≤   () and   () ≥ Δ + 1, where Δ represents the maximum degree of  [3].The Table 1 provides the existing results in graceful coloring.The graceful coloring of a few subclasses of unicyclic graphs [1] and a few variants of ladder graphs [7] are discussed in the literature.The Lemma 1.1 is useful in proving the main results of our paper, which was already proved in [7].
Complete bipartite graph of order 2  - [ 3] Trees with maximum degree Δ -Upper bound: Caterpillar with a maximum degree vertex adjacent to two vertices of maximum degree Rooted trees of height 2 Rooted trees of height 3 Rooted trees of height 4 Rooted trees with height at least 2 + ⌊ 1 3 Δ⌋ Graceful coloring for many graph classes like bipartite graphs, complete graphs, trees, etc. are still open.Hence we focus on a subclass of trees.

Our contributions:
For the trees with maximum degree 4, we -find the bound for the graceful chromatic number.
-give a structural characterization.
-provide an algorithm for graceful coloring.

Notations and terminologies
Let  represent the class of trees  with maximum degree 4 and the following notations hold for all  ∈  which is clearly illustrated in the Fig. 1.

Main results
It is proved that for any connected graph ,   () ≥ Δ + 1 and for a nontrivial tree  ,   ( ) ≤ ⌈ 5Δ 3 ⌉ where Δ is the maximum degree of the graph [3].Hence, we conclude that, 5 ≤   ( ) ≤ 7 for all  ∈ .In addition, we categorize  into three types based on the graceful chromatic number.
Let  be a root containing a neighbor  whose degree is 1.We now state few conditions to generate trees in  2 : Note that the trees that belong to  satisfying   , 1 ≤  ≤ 3 do not belong to  1 .
Thus, in all Cases   ( ) ≠ 5, for all  ∈  2 .□ Denote  1 as a tree in  containing a path  −  −  with () = () = 4 and () = 3 such that () = {, , } and () = 1.Also, denote  2 as a tree in  containing a path  −  −  with () = () = 4 and () = 3 such that () = {, , } and () = 1.We now define a new graph operation "Pendant vertex identification ()".: Consider two or more vertex disjoint graphs with at least one pendant vertex in each of the graphs.This graph operation merges the pendant vertices in each graph into a single vertex, preserving their adjacencies.We now construct  3 as the class of all trees in  which contain an induced subgraph by using (), () and ().(A) Applying the graph operation  for  1 with a tree from  1 or the tree  0 .(B) Applying the graph operation  for  1 and  2 .(C) Repeated application of  on  1 .Proposition 3.5.  ( ) ≠ 5, for all  ∈  3 .
Clearly the degree of the identified vertex  in