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.

1. Introduction

All graphs considered here are finite, undirected, and without loops and multiple edges. Let G be a graph with the vertex set V=V(G) and the edge set E=E(G). 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 vertex-set 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].

A bijection g:V(G)→{1,2,…,|V(G)|} is called (a,d)-edge-antimagic vertex ((a,d)-EAV) labeling of G if the set of edge-weights of all edges in G is equal to the set {a,a+d,a+2d,…,a+(|E(G)|-1)d}, where a>0 and d≥0 are two fixed integers. The edge-weight wg(xy) of an edge xy∈E(G) under the vertex labeling g is defined as the sum of the labels of its end vertices; that is, wg(xy)=g(x)+g(y). A graph that admits (a,d)-EAV labeling is called an (a,d)-EAV graph.

A bijection f: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 set of edge-weights of all edges in G forms an arithmetic progression starting from a with the difference d, where a>0 and d≥0 are two fixed integers. The edge-weight wf(xy) of an edge xy∈E(G) under the total labeling f is defined as the sum of the edge label and the labels of its end vertices. This means that{wf(xy)=f(x)+f(y)+f(xy):xy∈E(G)}={a,a+d,a+2d,…,a+(|E(G)|-1)d}. Moreover, if the vertices are labeled with the smallest possible numbers, that is, f(V)={1,2,…,|V(G)|}, then the labeling is called super (a,d)-edge-antimagic total (super (a,d)-EAT). A graph that admits an (a,d)-EAT labeling or a super (a,d)-EAT labeling is called an (a,d)-EAT graph or a super (a,d)-EAT graph, respectively.

The super (a,0)-EAT labelings are usually called super edge-magic; see [5–8]. Definition of super (a,d)-EAT labeling was introduced by Simanjuntak et al. [9]. This labeling is a natural extension of the notions of edge-magic labeling; see [7, 8]. Many other researchers investigated different types of antimagic graphs. For example, see Bodendiek and Walther [10], Hartsfield and Ringel [11].

In [9], Simanjuntak et al. defined the concept of (a,d)-EAV graphs and studied the properties of (a,d)-EAV labeling and (a,d)-EAT labeling and gave constructions of (a,d)-EAT labelings for cycles and paths. Bača et al. [12] presented some relation between (a,d)-EAT labeling and other labelings, namely, edge-magic vertex labeling and edge-magic total labeling.

In this paper, we study super (a,d)-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.

2. Basic Properties

Let us first recall the known upper bound for the parameter d of super (a,d)-EAT labeling. Assume that a graph G has a super (a,d)-EAT labeling f, f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|}. The minimum possible edge-weight in the labeling f is at least 1+2+(|V(G)|+1)=|V(G)|+4. Thus, a≥|V(G)|+4. On the other hand, the maximum possible edge-weight is at most (|V(G)|-1)+|V(G)|+(|V(G)|+|E(G)|)=3|V(G)|+|E(G)|-1. So
(1)a+(|E(G)|-1)d≤3|V(G)|+|E(G)|-1,(2)d≤2|V(G)|+|E(G)|-5|E(G)|-1.

Thus, we have the upper bound for the difference d. In particular, from (2), it follows that, for any connected graph, where |V(G)|-1≤|E(G)|, the feasible value d is no more than 3.

The next proposition, proved by Bača et al. [12], gives a method on how to extend an edge-antimagic vertex labeling to a super edge-antimagic total labeling.

Proposition 1 (see [<xref ref-type="bibr" rid="B2">12</xref>]).

If a graph G has an (a,d)-EAV labeling, then

Ghas a super (a+|V(G)|+1,d+1)-EAT labeling, and

G has a super (a+|V(G)|+|E(G)|,d-1)-EAT labeling.

The following lemma will be useful to obtain a super edge-antimagic total labeling.

Lemma 2 (see [<xref ref-type="bibr" rid="B17">13</xref>]).

Let 𝔄 be a sequence 𝔄={c,c+1,c+2,…,c+k}, k even. Then, there exists a permutation Π(𝔄) of the elements of 𝔄, such that 𝔄+Π(𝔄)={2c+k/2,2c+k/2+1,2c+k/2+2,…,2c+3k/2-1,2c+3k/2}.

Using this lemma, we obtain that, if G is an (a,1)-EAV graph with odd number of edges, then G is also super (a′,1)-EAT.

3. Crowns

If G has order n, the corona of G with H, denoted by G⊙H, is the graph obtained by taking one copy of G and n copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H. A cycle of order n with an m pendant edges attached at each vertex, that is, Cn⊙mK1, is called an m-crown with cycle of order n. 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 n 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 n.

Silaban and Sugeng [14] showed that, if the m-crown Cn⊙mK1 is (a,d)-EAT, then d≤5. They also describe (a,d)-EAT, labeling of the m-crown for d=2 and d=4. Note that the (a,2)-EAT labeling of m-crown presented in [14] is super (a,2)-EAT. Moreover, they proved that, if m≡1(mod4), n and d are odd; there is no (a,d)-EAT labeling for Cn⊙mK1. They also proposed the following open problem.

Open Problem 1 (see [<xref ref-type="bibr" rid="B15">14</xref>]).

Find if there is an (a,d)-EAT labeling, d∈{1,3,5} for m-crown graphs Cn⊙mK1.

Figueroa-Centeno et al. [15] proved that the m-crown graph has a super (a,0)-EAT labeling.

Proposition 3 (see [<xref ref-type="bibr" rid="B9">15</xref>]).

For every two integers n≥3 and m≥1, the m-crown Cn⊙mK1 is super (a,0)-EAT.

According to inequality (2), we have that, if the crown Cn⊙K1 is super (a,d)-EAT, then d≤2. Immediately from Proposition 3 and the results proved in [14], we have that the crown Cn⊙K1 is super (a,0)-EAT and super (a,2)-EAT for every positive integer n≥3. Moreover, for n odd, the crown is not (a,1)-EAT. In the following theorem, we prove that the crown is super (a,1)-EAT for n even. Thus, we partially give an answer to Open Problem 1.

Theorem 4.

For every even positive integer n, n≥4, the crown Cn⊙K1 is super (a,1)-EAT.

Proof.

Let V(Cn⊙K1)={v1,v2,…,vn,u1,u2,…,un} be the vertex set and E(Cn⊙K1)={v1v2,v2v3…,vnv1,v1u1,v2u2,…,vnun} be the edge set of Cn⊙K1; see Figure 1.

We define a labeling f, f:V(Cn⊙K1)∪E(Cn⊙K1)→{1,2,…,4n} as follows:
(3)f(vi)={i+n,fori=1,2,…,n2;i,fori=n2+1,n2+2,…,n;f(ui)={i,fori=1,2,…,n2;i+n,fori=n2+1,n2+2,…,n;f(vivi+1)={7n2+1-i,fori=1,2,…,n2;9n2+1-i,fori=n2+1,n2+2,…,n-1;f(v1vn)=7n2+1;f(viui)=3n+1-i,fori=1,2,…,n.

It is easy to check that f is a bijection. For the edge-weights under the labeling f, we have
(4)wf(v1vn)=11n2+2;wf(vivi+1)={11n2+2+i,fori=1,2,…,n2-1;9n2+2+i,fori=n2,n2+1,…,n-1;wf(viui)=4n+1+i,fori=1,2,…,n.
Thus, the edge-weights are distinct number from the set {4n+2,4n+3,…,6n+1}. This means that f is a super (a,1)-EAT labeling of the crown Cn⊙K1.

The crown Cn⊙K1.

In [16], was proved the following.

Proposition 5 (see [<xref ref-type="bibr" rid="B3">16</xref>]).

Let G be a super (a,1)-EAT graph. Then, the disjoint union of arbitrary number of copies of G, that is, sG, s≥1, also admits a super (a′,1)-EAT labeling.

In [17], Figueroa-Centeno et al. proved the following.

Proposition 6 (see [<xref ref-type="bibr" rid="B10">17</xref>]).

If G is a (super) (a,0)-EAT bipartite or tripartite graph and s is odd, then sG is (super) (a′,0)-EAT.

For (super) (a,2)-EAT labeling for disjoint union of copies of a graph, was shown the following.

Proposition 7 (see [<xref ref-type="bibr" rid="B5">18</xref>]).

If G is a (super) (a,2)-EAT bipartite or tripartite graph and s is odd, then sG is (super) (a′,2)-EAT.

Using the above mentioned results, we immediately obtain the following theorem.

Theorem 8.

Let n be an even positive integer, n≥4. Then, the disjoint union of arbitrary number of copies of the crown Cn⊙K1, that is, s(Cn⊙K1), s≥1, admits a super (a,1)-EAT labeling.

Since the crown Cn⊙K1 is either a bipartite graph, for n even, or a tripartite graph, for n odd, thus, we have the following.

Theorem 9.

Let n be a positive integer, n≥3. Then, the disjoint union of odd number of copies of the crown Cn⊙K1, that is, s(Cn⊙K1), s≥1, admits a super (a,d)-EAT labeling for d∈{0,2}.

According to these results we are able to give partially positive answers to the open problems listed in [19].

Open Problem 2 (see [<xref ref-type="bibr" rid="B1">19</xref>]).

For the graph s(Cn⊙mK1), sn even, and m≥1, determine if there is a super (a,d)-EAT labeling with d∈{0,2}.

Open Problem 3 (see [<xref ref-type="bibr" rid="B1">19</xref>]).

For the graph s(Cn⊙mK1), s odd, and n(m+1) even, determine if there is a super (a,1)-EAT labeling.

4. Graphs Related to Crown Graphs

Let us consider the graph obtained from a crown graph Cn⊙K1 by deleting one pendant edge.

Theorem 10.

For n odd, n≥3, the graph obtained from a crown graph Cn⊙K1 by removing a pendant edge is super (a,d)-EAT for d∈{0,1,2}.

Proof.

Let G be a graph obtained from a crown graph Cn⊙K1 by removing a pendant edge. Without loss of generality, we can assume that the removed edge is v1u1. Other edges and vertices we denote in the same manner as that of Theorem 4. Thus |V(G)|=2n-1 and |E(G)|=2n-1. It follows from (2) that d≤2.

Define a vertex labeling g as follows:
(5)g(vi)={i,fori=1,3,…,n;i+n-1,fori=2,4,…,n-1;g(ui)={i+n-1,fori=3,5,…,n;i,fori=2,4,…,n-1.
It is easy to see that labeling g is a bijection from the vertex set to the set {1,2,…,2n-1}. For the edge-weights under the labeling g, we have
(6)wg(v1vn)=1+n;wg(vivi+1)=2i+n,fori=1,2,…,n-1;wg(viui)=2i+n-1,fori=2,3,…,n.
Thus, the edge-weights are consecutive numbers n+1,n+2,…,3n-1. This means that g is the (a,1)-EAV labeling of G. According to Proposition 1, the labeling g can be extended to the super (a0,0)-EAT and the super (a2,2)-EAT labeling of G. Moreover, as the size of G is odd, |E(G)|=2n-1, and according to Lemma 2, we have that G is also super (a1,1)-EAT.

Note that we found some (a,1)-EAV labelings for the graphs obtained from a crown graph Cn⊙K1 by removing a pendant edge only for small size of n, where n is even. In Figure 2, there are depicted (a,1)-EAV labelings of these graphs for n=4,6,8. However, we propose the following conjecture.

The (a,1)-EAV labelings for the graphs obtained from a crown graph Cn⊙K1 by removing one pendant edge for n=4,6,8. The edge-weights under the corresponding labelings are depicted in italic.

Conjecture 11.

The graph obtained from a crown graph Cn⊙K1, n≥3, by removing a pendant edge is super (a,d)-EAT for d∈{0,1,2}.

Now, we will deal with the graph obtained from a crown graph Cn⊙K1 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 G with the (a,1)-EAV labeling g defined in the proof of Theorem 10. It is easy to see that the vertex un is labeled with the maximal vertex label, g(un)=2n-1. Also, the edge-weight of the edge vnun is the maximal possible, wg(vnun)=g(vn)+g(un)=(2n-1)+n=3n-1. Thus, it is possible to remove the edge vnun from the graph G; we denote the graph by Gr, and for the labeling g restricted to the graph Gr, we denote it by gr. Clearly, rr is (a,1)-EAV labeling of Gr. According to Proposition 1 from the labeling gr, we obtain the super (a0,0)-EAT and the super (a2,2)-EAT labeling of Gr. Note that, as the size of Gr is even, |E(Gr)|=2n-2, the labeling gr can not be extended to a super (a1,1)-EAT labeling of Gr.

Theorem 12.

For odd n, n≥3, the graph obtained from a crown graph Cn⊙K1 by removing two pendant edges at distance 1 is super (a,d)-EAT for d∈{0,2}.

Next, we show that, if we remove from a crown graph Cn⊙K1, n≥5, two pendant edges at distance 2, the resulting graph is super (a,d)-EAT for d∈{0,2}.

Theorem 13.

For odd n, n≥5, the graph obtained from a crown graph Cn⊙K1 by removing two pendant edges at distance 2 is super (a,d)-EAT for d∈{0,2}.

Proof.

Let G be a graph obtained from a crown graph Cn⊙K1 by removing two pendant edges at distance 2. Let V(G)={v1,v2,…, vn,u2,u3,…,un-2,un} be the vertex set and E(G)={v1v2,v2v3…, vnv1,v2u2,v3u3,…,vn-2un-2,vnun} be the edge set of G. Thus, |V(G)|=2n-2 and |E(G)|=2n-2. Using (2) gives d≤2.

Define a vertex labeling g in the following way:
(7)g(vi)={i+12,fori=1,3,…,n;n+i+12,fori=2,4,…,n-1;g(ui)={3n-4+i2,fori=3,5,…,n;2n+i2,fori=2,4,…,n-3.
It is not difficult to check that the labeling g is a bijection from the vertex set of G to the set {1,2,…,2n-2} and that the edge-weights under the labeling g are consecutive numbers (n+3)/2,(n+5)/2,…,(5n-3)/2. Thus g is the (a,1)-EAV labeling of G. According to Proposition 1, it is possible to extend the labeling g to the super (a0,0)-EAT and the super (a2,2)-EAT labelings of G.

Result in the following theorem is based on the Petersen Theorem.

Proposition 14 (Petersen Theorem).

Let G be a 2r-regular graph. Then, there exists a 2-factor in G.

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 2-factorization.

The construction in the following theorem allows to find a super (a,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.

Theorem 15.

Let G be a graph that arose from a 2r-regular graph H, r≥0, by adding k pendant edges to k different vertices of H, 0≤k≤|V(H)|. If k is an even integer, then the graph G is super (a,1)-EAT.

Proof.

Let G be a graph that arose from a 2r-regular graph H, r≥0, by adding k pendant edges to k different vertices of H, 0≤k≤|V(H)| and let m=|V(H)|-k. Thus, |V(G)|=2k+m, |E(G)|=k+r(k+m), and |E(H)|=r(k+m).

Let k be an even integer. Let us denote the pendant edges of G by symbols e1,e2,…,ek. We denote the vertices of G such that
(8)ei=vivk+m+i,i=1,2,…,k,
and moreover
(9){vk/2+1,vk/2+2,…,vk,vk+m+1,vk+m+2,…,v3k/2+m}⊂V(H).
We denote the remaining m vertices of V(H) arbitrarily by the symbols vk+1,vk+2,…,vk+m.

By the Petersen Theorem, there exists a 2-factorization of H. We denote the 2-factors by Fj, j=1,2,…,r. Without loss of generality, we can suppose that V(H)=V(Fj) for all j, j=1,2,…,r and E(G)=∪j=1rE(Fj)∪{e1,e2,…,ek}. Each factor Fj 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 ejout(vt) the unique outgoing arc that forms the vertex vt in the factor Fj, t=k/2+1,k/2+2,…,3k/2+m. Note that each edge is denoted by two symbols.

We define a total labeling f of G in the following way:
(10)f(vi)=i,fori=1,2,…,2k+m;f(ei)=3k+m+1-i,fori=1,2,…,k;f(ejout(vt))=7k2+m+1-f(vt)+j(k+m),fort=k2+1,k2+2,…,3k2+m,j=1,2,…,r.

It is easy to see that the vertices are labeled by the first 2k+m integers, the edges e1,e2,…,ek by the next k labels, and the edges of H by consecutive integers starting at 3k+m+1. Thus, f is a bijection V(G)∪E(G)→{1,2,…3k+m+r(k+m)}.

It is not difficult to verify that f is a super (4k+2m+2,1)-EAT labeling of G. For the weights of the edges ei, i=1,2,…,k, we have
(11)wf(ei)=f(vi)+f(vk+m+1)+f(ei)=i+(k+m+i)+(3k+m+1-i)=4k+2m+1+i.
Thus, the edge-weights are the numbers
(12)4k+2m+2,4k+2m+3,…,5k+2m+1.
For convenience, we denote by vs the unique vertex such that vtvs=ejout(vt) in Fj, where s∈{k/2+1,k/2+2,…,3k/2+m}. The weights of the edges in Fj, j=1,2,…,r, are
(13)wf(ejout(vt))=f(vt)+f(vs)+f(ejout(vt))=f(vt)+f(vs)+(7k2+m+1-f(vt)+j(k+m))=7k2+m+1+j(k+m)+f(vs)
for all t=(k/2)+1,(k/2)+2,…,(3k/2)+m, and j=1,2,…,r. Since Fj is a factor in H it holds {f(vs):vs∈Fj}={k/2+1,k/2+2,…,3k/2+m}. Hence, we have that the set of the edge-weights in the factor Fj is
(14){4k+m+2+j(k+m),4k+m+3+j(k+m),…,5k+2m+1+j(k+m)},
and thus, the set of all edge-weights in G under the labeling f is
(15){4k+2m+2,4k+2m+3,…,5k+2m+1+r(k+m)}.

We conclude the paper with the result that immediately follows from the previous theorem.

Corollary 16.

Let G be a graph obtained from a crown graph Cn⊙K1, n≥3, by deleting any k pendant edges, 0≤k≤n. If n-k is even, then G is a super (a,1)-EAT graph.

Acknowledgments

This research is partially supported by FAST-National University of Computer and Emerging Sciences, Peshawar, and Higher Education Commission of Pakistan. The research for this paper was also supported by Slovak VEGA Grant 1/0130/12.

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