The paper deals with methods of constructing square difference labeling of caterpillars and graphs derived from two operations: path union of cycles and disjoint union of stars. The existence of the square difference labeling of disjoint union of any SD graph with path is proved

Sherman, Z.

Algebra and Discrete Mathematics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

“adm-n2” — 2018/7/24 — 22:32 — page 294 — #132Algebra and Discrete Mathematics RESEARCH ARTICLEVolume 25 (2018). Number 2, pp. 294–302c© Journal “Algebra and Discrete Mathematics”Square difference labeling of some union anddisjoint union graphsZoya ShermanCommunicated by D. SimsonAbstract. The paper deals with methods of constructingsquare difference labeling of caterpillars and graphs derived fromtwo operations: path union of cycles and disjoint union of stars. Theexistence of the square difference labeling of disjoint union of anySD graph with path is proved.IntroductionResearch studies conducted by Ajitha et. al. in the field of numbertheory inspired them to create two new types of labelings: square sumlabeling and square difference labeling. For the first time, square differencelabeling was introduced to scientific world in 2012 [1]. Its authors provedthe existence of this labeling for such classes of graphs as paths, stars,cycles, complete graphs, bipartite graphs, friendship graphs, triangularsnakes. Tharmaraj and Sarasija identified new classes of graphs that havesquare difference labeling, such as: fan Fn, graph gn, middle graphs of pathand cycle, total graphs of path and cycle [2,3]. In addition, they proved thatthe disjoint union of path with such graphs as star, path, cycle, and otherconstructions of graphs have the square difference labeling. Shiama [4]focused on the particular case of the caterpillar and demonstrated theexistence of the square difference labeling for the graph Pn ⊙K1. These2010 MSC: 05C78.Key words and phrases: square difference labeling, the square difference graph,caterpillar, disjoint union of graphs.“adm-n2” — 2018/7/24 — 22:32 — page 295 — #133Z. Sherman 295labelings, as well as majority of others, are well presented in the reviewpublished by Gallian [5].In this paper, we prove the existence of the square difference labelingfor new types of graphs, which were obtained using disjoint union of starsand path union of cycles. In addition, we show that the caterpillar, andthe disjoint union of any SD graph and path are square difference graphs.1. DefinitionsWe consider finite undirected graphs without loops and multiple edges.Let G=(V,E) be a graph with vertex set V =V (G) and edge set E=E(G).We assume that |V (G)| = p, |E(G)| = q.Definition 1 ([1]). A graph G = (V (G), E(G)) with |V (G)| = p issaid to be a square difference graph if there exist a bijectionf : V (G) → {0, 1, 2, . . . , p− 1} such that the induced an injection functionf∗ : E(G) → N given by f∗(uv) = |[f(u)]2−[f(v)]2| for every edge uv∈Gand the function f is called square difference labeling of the graph G.Graph, allowing square difference labeling, is called square differencegraph or SD graph.For n copies of the graph G=(V,E), we introduce the definition of thepath union.Definition 2 ([6]). Let the graphs G1, G2, . . . , Gn(n > 2) are n copies ofcertain graph G. Then the graph obtained via connection of Gi vertexwith the corresponding Gi+1 vertex, for i = 1, 2, . . . , n− 1, with edge iscalled the path graphs union G1, G2, . . . , Gn.Definition 3 ([6]). Disjoint union of graphs G1 = (V1, E1), G2 = (V2, E2),. . . , Gm = (Vm, Em) is a graph G = G1 ∪G2 ∪ · · · ∪Gm with vertex setV = V1 ∪ V2 ∪ · · · ∪ Vm and edge set E = E1 ∪ E2 ∪ · · · ∪ Em, whereV1 ∩ V2 ∩ · · · ∩ Vm = ⊘.Definition 4 ([7]). Caterpillar is a tree that after removing its end-vertices (vertices with degree 1), turns into a path. The path is called atree trunk and end-vertices are leaves of the tree, which grow out of thetrunk vertices.Theorem 1 ([8]). The square difference of consecutive numbers equalsthe sum of these numbers.Theorem 2 ([8]). If n = x2 − y2, then n ≡ 0, 1, 3 (mod 4).“adm-n2” — 2018/7/24 — 22:32 — page 296 — #134296 Square difference labeling of some unionLet f be square difference labeling for graph G, generating edge label-ing f∗, then for each edge uv of the set E(G), the following condition is true:f∗(u, v) ≡ 0, 1, 3 (mod 4). Furthermore, if f(u) = 0, then f∗(u, v) ≡ 0, 1(mod 4).2. Square difference labeling of caterpillarsTheorem 3. Any caterpillar is a square difference graph.Proof. Let G be the caterpillar with |V (G)| = kn. We define V (P ) ={v1, v2, . . . , vk} as a set of vertices of the trunk of the graph G and{ui,1, ui,2, . . . , ui,ni} as a set of leaves, for i = 1, 2, . . . , k.Let us analyze the case when i = 1. We obtain a caterpillar, which isthe star K1,n1 . We define the vertex labeling f of star K1,n1 as follows:f(v1) = 0,f(u1,j) = j, for j = 1, 2, . . . , n1.Function f defines bijection from the set of vertices into set {0, 1, . . . , n1}and induces a different edge labels, according to Definition 1. Edge labelsform a set {1, 4, 9, . . . , n21}. So labeling f is a square difference label-ing K1,n1 .Further, assume that i = l and l > 1. For the vertices of the trunkv1, v2, . . . , vl we assign labels 0, 1, 2, . . . , l − 1; and u1,1, . . . , u1,n1 , u2,1,. . . , u2,n2 , ui,1, . . . , ui,ni form a set {l, l + 1, . . . , l + n1 − 1, l + n1, . . . ,l +m− 1}, for m being the number of leaves.Trunk edge labels and end-edge labels will form two sets of numbers{1, 3, 5, . . . , 2l − 3}, and {l2, (l + 1)2, . . . , (l +m − 1)2 − (l − 1)2}. As aresult, we assign labels to the vertices, using all the positive integers from0 to kn− 1.Edge labeling f∗, generated by labeling f , presents an injection ofE(G) onto a certain subset of positive integers. According to Definition 1,labeling f of the graph G is a square difference labeling.3. Square difference labeling of path and cycle unionTheorem 4. An arbitrary path union of n copies of the cycle C3 is asquare difference graph for any positive integer n.Proof. Let graph G be formed by the path union of n copies of the cycleC3 as shown in Figure 1.“adm-n2” — 2018/7/24 — 22:32 — page 297 — #135Z. Sherman 297Figure 1. A square difference labeling the path union 3C3.Let u10, u20,. . . ,un0 be isomorphic images of vertex u of cycle C3 selectedrandomly. V (G)={ui0, wi1, wi2} — a set of vertices of graph G, for i =1, 2, . . . , n. We define the vertex labeling f of graph G with graph order|V (G)| = 3n as follows:f(ui0) = i− 1,f(wi1) = n+ 2i− 2,f(wi2) = n+ 2i− 1,for i = 1, 2, . . . , n. Therefore, function f is bijection of vertex set of graph Gonto set of numbers {0, 1, 2, . . . , 3n− 1}. Then edge labeling f∗ generatedby the function f , as defined by the Definition 1, is:f∗(ui0, ui+10) = |[f(ui0)]2 − [f(ui+10)]2|, (1)for i = 1, 2, . . . , n− 1;f∗(wi1, wi2) = |[f(wi1)]2 − [f(wi2)]2|, (2)f∗(wi1, ui0) = |[f(wi1)]2 − [f(ui0)]2|, (3)f∗(wi2, ui0) = |[f(wi2)]2 − [f(ui0)]2|, (4)for i = 1, 2, . . . , n.We apply Theorem 5 to formulas (1) and (2), and obtainf∗(ui0, ui+10) = f(ui0) + f(ui+10), (5)f∗(wi1, wi2) = f(wi1) + f(wi2). (6)Edge labels calculated by formulas (5) and (6) form sets of numbersS1 = {1, 3, 5, . . . , 2n − 3}, S2 = {2n + 1, 2n + 5, 2n + 9, . . . , 6n − 3}.Meanwhile, edge labels obtained by formulas (3) and (4) form set S3 ={n2, (n+ 1)2, . . . , (2n− 1)(4n− 3), 2n(4n− 3)}.“adm-n2” — 2018/7/24 — 22:32 — page 298 — #136298 Square difference labeling of some unionLet us consider minimum number of n2 in set S3, and compare it withthe maximum number of 6n−3 from set S1∪S2. If n > 6, then n2 > 6n−3and elements of set S = S1 ∪ S2 ∪ S3 ⊂ N are different, where N is theset of positive integers. They form a set S = S1 ∪ S2 ∪ S3 ⊂ N . Thus,for n > 6 function f∗ is an injection of E(G) into the S set. Labeling f ,according to Definition 1, is a square difference labeling, while graph G isa SD graph.For n = 1, 2, 3, 4, 5 square difference labeling of an arbitrary pathunion of n copies of the cycle C3 is shown in Figure 2, 3.Figure 2. Square difference labeling the path union nC3, for n = 1, 2.Figure 3. Square difference labeling the path union nC3, for n = 3, 4, 5.4. Square difference labeling of the disjoint union ofgraphsTheorem 5. The disjoint union of graphs G1, G2, . . . , Gm of the familyof graphs Gi = Ki1,ni, for i = 1, 2, . . . ,m is a square difference graph forany positive integers m and ni.Proof. The graph G is obtained by disjoint union of stars K1,ni , for i =1, 2, . . . ,m. Let V (K1,ni) = {ui0, ui1, ui2, . . . , uini} be the set of vertices of“adm-n2” — 2018/7/24 — 22:32 — page 299 — #137Z. Sherman 299the graph G, for ni can be any value starting from 1. We assume that ui0is a central vertex of star K1,ni . We define the vertex labeling f of thegraph G with graph order |V (G)| = (n1+n2+ · · ·+nm)+m as following:f(ui0) = i− 1, (7)f(uij) ={m− 1 + j if i = 1,Σi−1l=1nl +m− 1 + j otherwise,(8)for i = 1, 2, . . . ,m, j = 1, 2, . . . , ni. Function f defines bijection from theset of vertices into a set of numbers {0, 1, 2, . . . , (n1 + n2 + · · · + nm) +m− 1}. At the same time it induces the following edge labels for the firststars K1,n1 :f∗(u11, u10) = |[f(u11)]2 − [f(u10)]2| = |m2 − 0| = m2,f∗(u12, u10) = |[f(u12)]2 − [f(u10)]2| = |(m+ 1)2 − 0| = (m+ 1)2,f∗(u13, u10) = |[f(u13)]2 − [f(u10)]2| = |(m+ 2)2 − 0| = (m+ 2)2,· · ·f∗(u1n1 , u10) = |[f(u1n1)]2 − [f(u10)]2| = |(m+n1−1)2−0| = (m+n1−1)2.The calculated edge labels form a set of different numbers S1 = {m2,(m+ 1)2, . . . , (m+ n1 − 1)2}. Also, the function f generates edge labelsfor the second star K1,n2 :f∗(u21, u20) = |[f(u21)]2 − [f(u20)]2| = |(m+ n1)2 − 1|= |1 + 2(m+ n1 − 1) + (m+ n1 − 1)2 − 1|= 2(m+ n1 − 1) + (m+ n1 − 1)2,f∗(u22, u20) = |[f(u22)]2 − [f(u20)]2| = |(m+ n1 + 1)2 − 1|= |1 + 2(m+ n1) + (m+ n1)2 − 1|= 2(m+ n1) + (m+ n1)2,. . .f∗(u2n2 , u20) = |[f(u2n2)]2 − [f(u20)]2| = |(m+ n1 + n2 − 1)2 − 1|= |1 + 2(m+ n1 + n2 − 2) + (m+ n1 + n2 − 2)2 − 1|= 2(m+ n1 + n2 − 2) + (m+ n1 + n2 − 2)2.Thus, we get a set of numbers S2 = {2(m + n1 − 1) + (m+ n1 − 1)2,2(m+ n1) + (m+ n1)2, . . . , 2(m+ n1 + n2 − 2) + (m+ n1 + n2 − 2)2}.Similar actions are performed for all the stars in the disjoint union.Thus, the edge labels for the latest star K1,nm , induced by function f , form“adm-n2” — 2018/7/24 — 22:32 — page 300 — #138300 Square difference labeling of some uniona set of numbers Sm = {2(n1 + n2 + · · ·+ nm−1 + 1)(m− 1) + (n1 + n2 +· · ·+nm−1+1)2, . . . , 2(n1+n2+ · · ·+nm)(m−1)+(n1+n2+ · · ·+nm)2}.Consequently, set S = S1 ∪ S2 ∪ · · · ∪ Sm ⊂ N , consisting of differentnumbers is formed. Therefore, function f∗, is the injection of E(G) intothe set of positive integers. Labeling f for the graph G, according toDefinition 1, is square difference labeling.In the case of graph mK1,n, formulas (7) and (8) can be written asfollowing:f(ui0) = i− 1, (9)f(uij) = n(i− 1) +m− 1 + j, (10)for i = 1, 2, . . . ,m, j = 1, 2, . . . , n.Example 1. We apply these formulas (9) and (10) to the graph 3K1,4,and get square difference labeling of the graph 3K1,4 as shown in Figure 4.Figure 4. Square difference labeling the graph 3K1,4.In the following theorem, we summarize some results obtained in thearticles [2,3] concerning the square difference labeling of the disjoint unionof paths or cycles, or other types of SD graphs with path.Theorem 6. The disjoint union of any SD graph G with path Pk, is thesquare difference graph for any k.Proof. Let us consider the disjoint union of G and path Pk. Let V (G) ={w1, w2, . . . , wl} be the set of vertices of graph G. We denote V (Pk) ={v1, v2, . . . , vk} the set of vertices of path Pk. Let f be a square differencelabeling of G. We define the vertex labeling f1 of graph G∪Pk, as follows:f1(vi) = i− 1 for i = 1, 2, . . . , k,f1(wj) = f(wj) + k for j = 1, 2, . . . , l.“adm-n2” — 2018/7/24 — 22:32 — page 301 — #139Z. Sherman 301Function f1 defines bijection from V (G ∪ Pk) onto the set {0, 1, 2, . . . ,k + l − 1}. Edge labels of path image Pk in G∪Pk, induced by labeling f1,form a set of numbers {1, 3, 5, . . . , 2k − 3}.Let wn, wm be any adjacent vertices of G with labels f(wn) = n,f(wm) = m, for m,n ∈ {0, 1, . . . , l − 1}, n 6= m and f∗ be edge labelingof G, induced by labeling f . Also, we assume n > m. Thenf∗(wn, wm) = |[f(wn)]2 − [f(wm)]2 = |n2 −m2| = n2 −m2.Let us consider edge labeling f∗1 of the graph G ∪ Pk, induced bylabeling f1 and find an edge label wnwm:f∗1 (wn, wm) = |[f∗1 (wn)]2 − [f∗1 (wm)]2= |(n+ k)2 − (m+ k)2|= |n2 + 2nk + k −m2 − 2mk − k2|= n2 −m2 + 2k(n−m)= f∗(wn, wm) + 2k(n−m).(11)Since 2k(n−m) > 2k−3 for ∀m,n ∈ {0, 1, . . . , l−1}, then f∗1 (wn, wm) >2k− 3. Consequently, the edge labeling of graph G∪Pk is different. So f∗1is an injection. According to Definition 1, f1 is square difference labelingof graph G.Additionally, using analytical description of labeling obtained in The-orems 4 and 5, we developed algorithms for square difference labeling ofcertain graph examples for path union of cycles and disjoint union of stars.Results verify our theoretical findings.5. ConclusionIn this paper the class of square difference graphs is expanded. Theexistence of the square difference labeling of the disjoin union of any SDgraph with path is proved. The methods developed for constructing asquare difference labeling for caterpillars and graphs derived from twooperations: a path union of cycles and disjoint union of stars, may be usedin further theoretical studies.References[1] V. Ajitha, K. L. Princy, V. Lokesha and P. S. Ranjini. On square difference Graphs.International Journal of Mathematical Combinatorics, 1(1):31–40, 2012.“adm-n2” — 2018/7/24 — 22:32 — page 302 — #140302 Square difference labeling of some union[2] T. Tharmaraj, P. B. Sarasija. Square difference Labeling for Certain Graphs.International Journal of Mathematical Archive, 4(8):183–186, 2013.[3] T. Tharmaraj, P. B. Sarasija. Square difference Labeling of Some Union Graphs.International Journal of Mathematics Trends and Technology, 11(11):81–88, 2014.[4] J. Shiama. Square difference labeling for some path, fan and gear graphs. Interna-tional Journal of Scientific and Engineering Research, 4:1–9, 2013.[5] J. A. Gallian. A dynamic survey of graph labeling. The Electronic Journal ofCombinatorics, 18:#DS6, 2011.[6] F. Harary. Graph Theory. Addison Wesley, Reading, Massachusetts, 1969.[7] G. A. Donets, D. A. Petrenko. Building T-factorizations of the complete graphand the problem Rosa. International Journal of Control systems and machines,4:21–24, 2010.[8] S. G. Telang. Number Theory. (Sixth edition), TATA McGRAW-HILL, 1996.Contact informationZ. Sherman V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of Ukraine,Kyiv, UkraineE-Mail(s): sherman.zoya@gmail.comReceived by the editors: 02.09.2016and in final form 06.11.2016.

Square difference labeling of some union and disjoint union graphs

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Square difference labeling of some union and disjoint union graphs

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