On Super ( a , d )-Edge-Antimagic Total Labeling of Special Types of Crown Graphs

For a graph G = (V, E), a bijection f from V(G) ∪ E(G) → {1, 2, . . . , |V(G)| + |E(G)|} is called (a, d)-edge-antimagic total ((a, d)EAT) labeling of G if the edge-weights w(xy) = f(x) + f(y) + f(xy), xy ∈ E(G), form an arithmetic progression starting from a and having a common difference d, where a > 0 and d ≥ 0 are two fixed integers. An (a, d)-EAT labeling is called super (a, d)-EAT labeling if the vertices are labeled with the smallest possible numbers; that is,f(V) = {1, 2, . . . , |V(G)|}. In this paper, we study super (a, d)-EAT labeling of cycles with some pendant edges attached to different vertices of the cycle.


Introduction
All graphs considered here are finite, undirected, and without loops and multiple edges.Let  be a graph with the vertex set  = () and the edge set  = ().For a general reference of the graph theoretic notions, see [1,2].
A labeling (or valuation) of a graph is a map that carries graph elements to numbers, usually to positive or nonnegative integers.In this paper, the domain of the map is the set of all vertices and all edges of a graph.Such type of labeling is called total labeling.Some labelings use the vertexset only, or the edge-set only, and we will call them vertex labelings or edge labelings, respectively.The most complete recent survey of graph labelings can be seen in [3,4].
In this paper, we study super (, )-EAT labeling of the class of graphs that can be obtained from a cycle by attaching some pendant edges to different vertices of the cycle.

Basic Properties
Thus, we have the upper bound for the difference .In particular, from (2), it follows that, for any connected graph, where |()| − 1 ≤ |()|, the feasible value  is no more than 3.
Using this lemma, we obtain that, if  is an (, 1)-EAV graph with odd number of edges, then  is also super (  , 1)-EAT.

Crowns
If  has order , the corona of  with , denoted by  ⊙ , is the graph obtained by taking one copy of  and  copies of  and joining the th vertex of  with an edge to every vertex in the th copy of .A cycle of order  with an  pendant edges attached at each vertex, that is,   ⊙  1 , is called an -crown with cycle of order .A 1-crown, or only crown, is a cycle with exactly one pendant edge attached at each vertex of the cycle.For the sake of brevity, we will refer to the crown with cycle of order  simply as the crown if its cycle order is clear from the context.Note that a crown is also known in the literature as a sun graph.In this section, we will deal with the graphs related to 1-crown with cycle of order .
According to inequality (2), we have that, if the crown   ⊙  1 is super (, )-EAT, then  ≤ 2. Immediately from Proposition 3 and the results proved in [14], we have that the crown   ⊙  1 is super (, 0)-EAT and super (, 2)-EAT for every positive integer  ≥ 3.Moreover, for  odd, the crown is not (, 1)-EAT.In the following theorem, we prove that the crown is super (, 1)-EAT for  even.Thus, we partially give an answer to Open Problem 1. Theorem 4. For every even positive integer ,  ≥ 4, the crown   ⊙  1 is super (, 1)-EAT.
In [16], was proved the following.
For (super) (, 2)-EAT labeling for disjoint union of copies of a graph, was shown the following.
Using the above mentioned results, we immediately obtain the following theorem.Theorem 8. Let  be an even positive integer,  ≥ 4.Then, the disjoint union of arbitrary number of copies of the crown   ⊙  1 , that is, (  ⊙  1 ),  ≥ 1, admits a super (, 1)-EAT labeling.
Since the crown   ⊙  1 is either a bipartite graph, for  even, or a tripartite graph, for  odd, thus, we have the following.
According to these results we are able to give partially positive answers to the open problems listed in [19].

Graphs Related to Crown Graphs
Let us consider the graph obtained from a crown graph   ⊙  1 by deleting one pendant edge.
Proof.Let  be a graph obtained from a crown graph   ⊙  1 by removing a pendant edge.Without loss of generality, we can assume that the removed edge is V 1  1 .Other edges and vertices we denote in the same manner as that of Theorem ( It is easy to see that labeling  is a bijection from the vertex set to the set {1, 2, . . ., 2 − 1}.For the edge-weights under the labeling , we have Thus, the edge-weights are consecutive numbers  + 1,  + 2, . . ., 3 − 1.This means that  is the (, 1)-EAV labeling of .According to Proposition 1, the labeling  can be extended to the super ( 0 , 0)-EAT and the super ( 2 , 2)-EAT labeling of .Moreover, as the size of  is odd, |()| = 2 − 1, and according to Lemma 2, we have that  is also super ( 1 , 1)-EAT.
Note that we found some (, 1)-EAV labelings for the graphs obtained from a crown graph   ⊙  1 by removing a pendant edge only for small size of , where  is even.In Figure 2, there are depicted (, 1)-EAV labelings of these graphs for  = 4, 6, 8.However, we propose the following conjecture.
Conjecture 11.The graph obtained from a crown graph   ⊙  1 ,  ≥ 3, by removing a pendant edge is super (, )-EAT for  ∈ {0, 1, 2}.Now, we will deal with the graph obtained from a crown graph   ⊙  1 by removing two pendant edges at distance 1 and at distance 2.
First, consider the case when we remove two pendant edges at distance 1.Let us consider the graph  with the (, 1)-EAV labeling  defined in the proof of Theorem 10.It is easy to see that the vertex   is labeled with the maximal vertex label, (  ) = 2 − 1.Also, the edge-weight of the edge V    is the maximal possible,   (V    ) = (V  ) + (  ) = (2 − 1) +  = 3 − 1.Thus, it is possible to remove the edge V    from the graph ; we denote the graph by   , and for the labeling  restricted to the graph   , we denote it by   .Clearly,   is (, 1)-EAV labeling of   .According to Proposition 1 from the labeling   , we obtain the super ( 0 , 0)-EAT and the super ( 2 , 2)-EAT labeling of   .Note that, as the size of   is even, |(  )| = 2 − 2, the labeling   can not be extended to a super ( 1 , 1)-EAT labeling of   .Theorem 12.For odd ,  ≥ 3, the graph obtained from a crown graph   ⊙ 1 by removing two pendant edges at distance 1 is super (, )-EAT for  ∈ {0, 2}.
Result in the following theorem is based on the Petersen Theorem.
Notice that, after removing edges of the 2-factor guaranteed by the Petersen Theorem, we have again an even regular graph.Thus, by induction, an even regular graph has a 2factorization.
The construction in the following theorem allows to find a super (, 1)-EAT labeling of any graph that arose from an even regular graph by adding even number of pendant edges to different vertices of the original graph.Notice that the construction does not require the graph to be connected.Let  be an even integer.Let us denote the pendant edges of  by symbols  1 ,  2 , . . .,   .We denote the vertices of  such that and moreover We denote the remaining  vertices of () arbitrarily by the symbols V +1 , V +2 , . . ., V + .By the Petersen Theorem, there exists a 2-factorization of .We denote the 2-factors by   ,  = 1, 2, . . ., .Without loss of generality, we can suppose that () = (  ) for all ,  = 1, 2, . . .,  and () = ∪  =1 (  ) ∪ { 1 ,  2 , . . .,   }.Each factor   is a collection of cycles.We order and orient the cycles arbitrarily such that the arcs form oriented cycles.Now, we denote by the symbol  out  (V  ) the unique outgoing arc that forms the vertex V  in the factor   ,  = /2 + 1, /2 + 2, . . ., 3/2 + .Note that each edge is denoted by two symbols.
We define a total labeling  of  in the following way: It is easy to see that the vertices are labeled by the first 2+  integers, the edges  1 ,  2 , . . .,   by the next  labels, and the edges of  by consecutive integers starting at 3++1.Thus,  is a bijection () ∪ () → {1, 2, . . .3 +  + ( + )}.( For convenience, we denote by V  the unique vertex such that We conclude the paper with the result that immediately follows from the previous theorem.

Figure 2 :
Figure 2: The (, 1)-EAV labelings for the graphs obtained from a crown graph   ⊙  1 by removing one pendant edge for  = 4, 6, 8.The edge-weights under the corresponding labelings are depicted in italic.