A Conjecture on Super Edge-Magic Total Labeling of 4-Cycle Books

<jats:p>A graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> is called cycle books <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>B</mi>
                        <mfenced open="[" close="]" separators="|">
                           <mrow>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mn>4</mn>
                                    <mo>,</mo>
                                    <mi>m</mi>
                                 </mrow>
                              </mfenced>
                              <mo>,</mo>
                              <mn>2</mn>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> consists of m cycles <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <msub>
                           <mrow>
                              <mi>C</mi>
                           </mrow>
                           <mrow>
                              <mn>4</mn>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula> with a common path <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <msub>
                           <mrow>
                              <mi>P</mi>
                           </mrow>
                           <mrow>
                              <mn>2</mn>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula>. Figueroa-Centeno, Ichishima, and Muntaner-Batle conjecture that the graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>B</mi>
                        <mfenced open="[" close="]" separators="|">
                           <mrow>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mn>4</mn>
                                    <mo>,</mo>
                                    <mi>m</mi>
                                 </mrow>
                              </mfenced>
                              <mo>,</mo>
                              <mn>2</mn>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> is super edge-magic total if and only if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>m</mi>
                     </math>
                  </jats:inline-formula> is even or <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>m</mi>
                        <mo>≡</mo>
                        <mn>5</mn>
                        <mtext> </mtext>
                        <mi mathvariant="normal">mod</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mn>8</mn>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>. In this article, we prove this conjecture for <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mi>m</mi>
                     </math>
                  </jats:inline-formula> ≥ 36 and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
                        <mi>m</mi>
                     </math>
                  </jats:inline-formula>  = 0 mod (2).</jats:p>


Introduction
For undefined terms and notations in this study, we follow Chartrand, Lesniak, and Peng [1]. Let G be a graph with V(G) and E(G) be a set of vertices and edges, respectively. A graph G is called a (p, q) graph if G has p and q number of vertices and edges, respectively. Kotzig and Rosa [2] defined that an edge-magic total labeling of G is a bijective function f: V(G) ∪ E(G) ⟶ {1, 2, ..., p + q}, such that f(w) + f(wz) + f(z) � k for any edge wz ∈ E(G). Moreover, a super edgemagic total labeling is an edge-magic total labeling f, such that f(V(G)) � {1, 2, ..., p}. e notion of edge-magic total labeling of a graph is generalized to edge-antimagic total labeling of graphs. Let α ≥ 0 and β ≥ 0 be integers. Let W � w(xy): w(xy) � f (x) + f(xy) + f(y), xyE(G)}. If W forms an arithmetic sequence starting from α with common difference β, then G is called (α, β) − edge antimagic total labeling. Moreover, if f(V(G)) � {1, 2, ..., p}, then G is called super(α, β) − edge antimagic total labeling. Notice that when β � 0, the (α, β) − edge antimagic total labeling of G is the usual edgemagic total labeling of G with f(w) + f(wz) + f(z) � α for any edge wz ∈ E(G).
One of the most popular problems in the theory of graph labeling is super edge-magic total labeling of tree. Enomoto et al. [3] proposed the following conjecture.
Tree conjecture [3]: every tree is super(α, 0)− edge antimagic total labeling. e tree conjecture is still an open problem; however, some authors proved that tree conjecture is true for some classes of tree. For example, Bhatti, Javaid and Hussain [4] and Raheem et al. [5] proved that tree conjecture is true for subdivision of caterpillar. Javaid, Bhatti, and Aslam [6] proved that tree conjecture is true for subdivision of stars. Other authors who studied tree conjecture can be found in Gallian [7].
Another popular problem in the theory of graph labeling is super edge-magic total labeling of a cycle book. A cycle book graph is constructed from some cycles either with the same or different order. Let m ≥ 1 be any positive integer and let C a be the cycles of order a. Graph B[(a, m), 2] is constructed from some cycles C a of the same order. Swita et al. [9] contructed a graph from some cycles with different orders. A graph (a, b) − cycle book B [(a, m), (b, n), t] is constructed from some cycles C a and C b with a common path P t , a path of order t with m, n, a, b, and t as the positive integers.
Problem 2. Are all graphs B[(a, m), (b, n), 2] edge-magic total (super edge-magic total)? Both Problems 1 and 2 are interesting problems for at least the following two reasons. First reason is the solutions of Problems 1 and 2 that can be used to construct the secret sharing scheme in information technology. Reddy and Basha [10] and Imron et al. [11] used edge-magic total labeling of catepilar graphs to construct the secret sharing scheme. Baskoro, Simanjuntak, and Adithia [12] used edge-magic total labeling of star graphs to construct the secret sharing scheme. e second reason is both Problems 1 and 2 provide a challenging problem for the researchers, since they are open problems. Swita et al. [9] proved Problem 2 for a � 7 or a � 4x − 1 for any integer x. MacDougall and Wallis [13] proved Problem 2 for m � n � 1 that a graph B[(a, 1), (b, 1), 2] is a super edge-magic total labeling. Let l � min(a, b) − 3. Notice that l is a chord of cycle C (a+b− 2) . us, l is a chord of graph B[(a, 1), (b, 1), 2]. Using Kotzig array, Singgih [14,15] proposed a new method to construct an edge-magic total labeling (super edge-magic) of graph cycle C (a+b− 2)(2r+1) h with [(2r + 1) h ]z chords, each of length l � (a, b) − 3, from an edge-magic total labeling (super edge-magic) of graph B[(a, 1), (b, 1), 2], where hand z are the positive integers.
Cycle book conjecture [18]: the graph B[(4, m), 2] is super edge-magic total if and only if m is even or m ≡ 5 mod(8).
Gallian [7] reported that Yuansheng et al. [19] proved this conjecture for m is even in Ars Combinatoria, 93 (2009) 431-438. A study [20] contains the abstract of Yuansheng et al. [19] and claims that Yuansheng et al. proved the cycle book conjecture is true for m is even. e study [19] is the same as that of Gallian [7]. We trace this reference, and we find that this reference is neither in the  (2). e solution of cycle book conjecture is available from the author for 12 ≤ m ≤ 34 and m � 0 mod (2).

Preliminary Notes
In this section, we provide some previous results on super edge-magic total labeling of a graph. Figuero-Centeno, Ichisma, and Mutaner-Batle [18] proved some necessary conditions for super edge-magic total labeling of a graph. We need them to prove the main results of this study. First, we define some notations in the following definition. Theorem 1 (see [18]). Let G be a graph, such that | (VG)| � p and |E(G)| � q. en, G is super edge-magic total if and only if there exists a bijective function f: Theorem 2 (see [18]). Let G be a graph, such that | (VG)| � p and |E(G)| � q and f be a super edge-magic total labeling of Theorem 3 (see [18]). Let G be a graph B[(4, m), 2], such that S � f(w) + f(z): wz ∈ E(G) and s � min(S). If G is super edge-magic total labeling, then s � m/2 + 3. e following theorem is derived from the proof of eorem 3 in [3]. For self-contained of this article, we rewrite the proof again.

Theorem 4 (see [18]). Let G be a graph B[(4, m), 2] in Definition 1 and let
We substitute q � 3m + 1 to the last equation, and we have the following equation.

Proof of Cycle Book Conjecture for m Is Even
In this section, we prove that the cycle book conjecture is true if m is even and m ≥ 36.
and let G be a super edge-magic total. Note that p � 2m + 2 and q � 3m + 1. Let f be an edge-magic total labeling of G. If f is a super edge-magic total labeling of G, then the conditions (i) and (ii) follow from eorems 4 and 3, respectively, and the conditions (iii) and (iv) follow from eorem 1.