Jeyanthi P.

P. Jeyanthi

P. Selvagopal

Selvagopal P.

Jeyanthi, P.

1986

Selvagopal, P.

1987

Tokyo University of Science Repository for Academic Resources / 東京理科大学学術リポジトリ

SUT Journal of MathematicsVol. 44, No. 1 (2008), 23–31Magic covering of chain of an arbitrary 2-connectedsimple graphP. Jeyanthi and P. Selvagopal(Received August 19, 2007; Revised November 30, 2007)Abstract. A simple graph G = (V,E) admits an H-covering if every edgein E belongs to a subgraph of G isomorphic to H. We say that G is H-magicif there is a total labeling f : V ∪ E → {1, 2, 3, . . . , |V | + |E|} such that foreach subgraph H ′ = (V ′, E′) of G isomorphic to H,Pv∈V ′ f(v) +Pe∈E′ f(e)is constant. When f(V ) = {1, 2, . . . , |V |}, then G is said to be H-supermagic.In this paper we show that a chain of any 2-connected simple graph H is H-supermagic.AMS 2000 Mathematics Subject Classification. 20J06.Key words and phrases. Chain of graph, Magic covering and H-supermagic.§1. IntroductionThe concept of H-magic graphs was introduced in [2]. An edge-covering of agraph G is a family of different subgraphs H1, H2, . . . , Hk such that each edgeof E belongs to at least one of the subgraphs Hi, 1 ≤ i ≤ k. Then, it is saidthat G admits an (H1,H2, . . . ,Hk)-edge covering. If every Hi is isomorphic toa given graph H, then we say that G admits an H-covering.Suppose that G = (V,E) admits an H-covering. We say that a bijectivefunction f : V ∪ E → {1, 2, 3, . . . , |V | + |E|} is an H-magic labeling of G ifthere is a positive integer m(f), which we call magic sum, such that for eachsubgraph H ′ = (V ′, E′) of G isomorphic to H, we have, f(H ′) =∑v∈V ′ f(v)+∑e∈E′ f(e) = m(f). In this case we say that the graph G is H-magic. Whenf(V ) = {1, 2, . . . , |V |}, we say that G is H-supermagic and we denote itssupermagic-sum by s(f).We use the following notations. For any two integers n < m, we denote by[n,m], the set of all consecutive integers from n to m. For any set I ⊂ N wewrite∑I =∑x∈I x and for any integers k, I + k = {x + k : x ∈ I}. Thus2324 P. JEYANTHI AND P. SELVAGOPALk + [n,m] is the set of consecutive integers from k + n to k +m. It can beeasily verified that∑(I+ k) =∑I+ k|I|. Finally, given a graph G = (V,E)and a total labeling f on it we denote by f(G) =∑f(V ) +∑f(E).In [2], A. Gutierrez, and A. Llado studied the families of complete andcomplete bipartite graphs with respect to the star-magic and star-supermagicproperties and proved the following results.• The star K1,n is K1,h-supermagic for any 1≤ h ≤ n.• Let G be a d-regular graph. Then G is not K1,h-magic for any 1 < h < d.• (a) The complete graph Kn is not K1,h-magic for any 1 < h < n− 1.(b) The complete bipartite graph Kn,n is not K1,h-magic for any 1 <h < n.• The complete bipartite graph Kn,n is K1,n-magic for n ≥ 1.• The complete bipartite graphKn,n is notK1,n-supermagic for any integern > 1.• For any pair of integers 1 < r < s, the complete bipartite graph Kr,s isK1,h-supermagic if and only if h = s.The following results regarding path-magic and path-supermagic coveringsare also proved in [2].• The path Pn is Ph-supermagic for any integer 2 ≤ h ≤ n.• Let G be a Ph-magic graph, h > 2. Then G is Ch-free.• The complete graph Kn is not Ph-magic for any 2 < h ≤ n.• The cycle Cn is Ph-supermagic for any integer 2 ≤ h < n such thatgcd (n, h(h− 1)) = 1.Also in [2], the authors constructed some families of H-magic graphs for agiven graph H by proving the following results.• Let H be any graph with |V (H)|+ |E(H)| even. Then the disjoint unionG = kH of k copies of H is H-magic.Let G and H be two graphs and e ∈ E(H) a distinguished edge in H. Wedenote by G ∗ eH the graph obtained from G by gluing a copy of H to eachedge of G by the distinguished edge e ∈ E(H).• Let H be a 2-connected graph and G an H-free supermagic graph. Let kbe the size of G and h = |V (H)|+ |E(H)|. Assume that h and k are notboth even. Then, for each edge e ∈ E(H), the graph G∗ eH is H-magic.MAGIC COVERING OF A CHAIN OF 2-CONNECTED GRAPH 25In [3], P. Selvagopal and P. Jeyanthi proved that for any positive integer n,k- polygonal snake of length n is Ck-supermagic.In this paper we construct a chain graph Hn of 2-connected graph H oflength n, and prove that a chain graph Hn is H-supermagic.§2. Preliminary ResultsLet P = {X1, X2, . . . , Xk} be partition set of a set X of integers. When all setshave the same cardinality we say then P is a k-equipartition of X. We denotethe set of subsets sums of the parts of P by∑P = {∑X1,∑X2, . . . ,∑Xk}.The following lemmas are proved in [2].Lemma 1. Let h and k be two positive integers and let n = hk. For eachinteger 0 ≤ t ≤ ⌊h2⌋ there is a k-equipartition P of [1, n] such that ∑P is anarithmetic progression of difference d = h− 2t.Lemma 2. Let h and k be two positive integers and let n = hk. In the twofollowing cases there exists a k-equipartition P of a set X such that∑P is aset of consecutive integers.(i) h or k are not both even and X = [1, hk](ii) h = 2 and k is even and X = [1, hk + 1]− {k2 + 1}.We have the following four results from the above two lemmas.(a) If h is odd, then there exists a k-equipartition P = {X1, X2, . . . , Xk} ofX = [1, hk] such that∑P is a set of consecutive integers and∑P =(h−1)(hk+k+1)2 + [1, k].(b) If h is even, then there exists a k-equipartition P = {X1, X2, . . . , Xk} ofX = [1, hk] such that subsets sum are equal and is equal to h(hk+1)2 .(c) If h is even and k is odd, then there exists a k-equipartitionP = {X1, X2, . . . , Xk} of X = [1, hk] such that∑P is a set of consecutiveintegers and∑P = h(hk+1)2 +[−k−12 , k−12 ].(d) If h = 2 and k is even, and X = [1, 2k + 1] − {k2 + 1} then there existsa k-equipartition P = {X1, X2, . . . , Xk} of X such that∑P is a set ofconsecutive integers and∑P =[3k2 + 3,5k2 + 2].We generalise the second part of Lemma 2.Corollary 1. Let h and k be two even positive integers and h ≥ 4. If X =[1, hk + 1] − {k2 + 1}, there exists a k-equipartition P of X such that ∑P isa set of consecutive integers.26 P. JEYANTHI AND P. SELVAGOPALProof. Let Y = [1, 2k + 1] − {k2 + 1} and Z = (2k + 1) + [1, (h− 2)k]. ThenX = Y ∪ Z. By (d), there exists a k-equipartition P1 = {Y1, Y2, . . . , Yk} of Ysuch that ∑P1 =[3k2+ 3,5k2+ 2].As h−2 is even, by (b) there exists a k-equipartition P ′2 = {Z ′1, Z ′2, . . . , Z ′k}of [1, (h− 2)k] such that∑P ′2 ={(h− 2)(hk − 2k + 1)2}.Hence, there exists a k-equipartition P2 = {Z1, Z2, . . . , Zk} of Z such that∑P2 ={(h− 2)(2k + 1) + (h− 2)(hk − 2k + 1)2}.Let Xi = Yi ∪ Zi for 1 ≤ i ≤ k. Then P = {X1, X2, . . . , Xk} is a k-equipartition of X such that∑P is a set of consecutive integers and∑P = (h− 2)(2k + 1) + (h− 2)(hk − 2k + 1)2+[3k2+ 3,5k2+ 2].§3. Chain of an arbitrary simple connected graphLet H1,H2, . . . ,Hn be copies of a graph H. Let ui and vi be two distinctvertices of Hi for i = 1, 2, . . . , n. We construct a chain graph Hn of H oflength n by identifying two vertices ui and vi+1 for i = 1, 2, . . . , n − 1. SeeFigures 1 and 2.§4. Main ResultTheorem 1. Let H be a 2-connected (p, q) simple graph. Then Hn is H-supermagic if any one of the following conditions is satisfied.(i) p+ q is even(ii) p+ q + n is evenProof. Let G = (V,E) be a chain of n copies of H. Let us denote the ith copyof H in Hn by Hi = (Vi, Ei). Note that |V | = np − n + 1 and |E| = nq.Moreover, we remark that by H is a 2-connected graph, Hn does not containa subgraph H other than Hi.MAGIC COVERING OF A CHAIN OF 2-CONNECTED GRAPH 27Let vi be the vertex in common with Hi and Hi+1 for 1 ≤ i ≤ n − 1. Letv0 and vn be any two vertices in H1 and Hn respectively so that v0 6= v1 andvn 6= vn−1. Let V ′i = Vi − {vi−1, vi} for 1 ≤ i ≤ n.Case (i): p+ q is evenSuppose p and q are odd. As p − 2 is odd, by (a) there exists an n-equipartition P ′1 = {X ′1, X ′2, . . . , X ′n} of [1, n(p− 2)] such that∑P ′1 =(p− 3)(np− n+ 1)2+ [1, n] .Adding n+1 to [1, n(p− 2)], we get an n-equipartition P1 = {X1, X2, . . . , Xn}of [n+ 2, np− n+ 1] such that∑P1 = (p− 2)(n+ 1) + (p− 3)(np− n+ 1)2 + [1, n]Similarly, since q is odd there exists an n-equipartition P2 = {Y1, Y2, . . . , Yn}of (np− n+ 1) + [1, nq] such that∑P2 = q(np− n+ 1) + (q − 1)(nq + n+ 1)2 + [1, n]Define a total labeling f : V ∪ E → {1, 2, 3, . . . , np+ nq − n+ 1} as follows:(i) f(vi) = i+ 1 for 0 ≤ i ≤ n.(ii) f(V ′i ) = Xn−i+1 for 1 ≤ i ≤ n.(iii) f(Ei) = Yn−i+1 for 1 ≤ i ≤ n.Then for 1 ≤ i ≤ n,f(Hi) = f(vi−1) + f(vi) +∑f(V ′i ) +∑f(Ei)= f(vi−1) + f(vi) +∑Xn−i+1 +∑Yn−i+1=n(p+ q)2 + 3(p+ q)− 2n(p+ q) + 2n− 22As Hi ∼= H for 1 ≤ i ≤ n, Hn is H-supermagic.Suppose both p and q are even. As p is even, by Lemma 1, there existsan n-equipartition P ′1 = {X ′1, X ′2, . . . , X ′n} of [1, n(p− 2)] such that∑P ′1 isarithmetic progression of difference 2 and∑P ′1 ={n[(p− 2)2 − 2]+ p− 42+ 2r : 1 ≤ r ≤ n}.28 P. JEYANTHI AND P. SELVAGOPALAdding n+1 to [1, n(p− 2)], we get an n-equipartition P1 = {X1, X2, . . . , Xn}of [n+ 2, np− n+ 1] such that∑P1 ={(p− 2)(n+ 1) + n[(p− 2)2 − 2]+ p− 42+ 2i : 1 ≤ i ≤ n}As q is even, by (b), there exists an n-equipartition P ′2 = {Y ′1 , Y ′2 , . . . , Y ′n}of [1, nq] such that∑P ′2 ={q(nq+1)2}.Adding np− n+ 1 to [1, nq] there exists an n-equipartitionP2 = {Y1, Y2, . . . , Yn} of (np− n+ 1) + [1, nq] such that∑P2 ={q(np− n+ 1) + q(nq + 1)2}Define a total labeling f : V ∪ E → {1, 2, 3, . . . , np+ nq − n+ 1} as follows:(i) f(vi) = i+ 1 for 0 ≤ i ≤ n.(ii) f(V ′i ) = Xn−i+1 for 1 ≤ i ≤ n.(iii) f(Ei) = Yn−i+1 for 1 ≤ i ≤ n.Then for 1 ≤ i ≤ n,f(Hi) = f(vi−1) + f(vi) +∑f(V ′i ) +∑f(Ei)= f(vi−1) + f(vi) +∑Xn−i+1 +∑Yn−i+1=n(p+ q)2 + 3(p+ q)− 2n(p+ q) + 2n− 22As Hi ∼= H for 1 ≤ i ≤ n, Hn is H-supermagic.Case (ii): p + q + n is even: Suppose p is odd, q is even and n is odd.Since p is odd as in proof of Case (i), there exists an n-equipartition P1 ={X1, X2, . . . , Xn} of [n+ 2, np− n+ 1] such that∑P1 = (p− 2)(n+ 1) + (p− 3)(np− n+ 1)2 + [1, n]Since q is even and n is odd, by (c) there exists an n-equipartition P ′2 ={Y ′1 , Y ′2 , . . . , Y ′n} of [1, nq] such that∑P ′2 =q(nq + 1)2+[−n− 12,n− 12].Adding np−n+1 to [1, nq] there exists an n-equipartition P2 = {Y1, Y2, . . . , Yn}of (np− n+ 1) + [1, nq] such that∑P2 = q(np− n+ 1) + q(nq + 1)2 +[−n− 12,n− 12]Define a total labeling f : V ∪ E → {1, 2, 3, . . . , np+ nq − n+ 1} as follows:MAGIC COVERING OF A CHAIN OF 2-CONNECTED GRAPH 29(i) f(vi) = i+ 1 for 0 ≤ i ≤ n.(ii) f(V ′i ) = Xn−i+1 for 1 ≤ i ≤ n.(iii) f(Ei) = Yn−i+1 for 1 ≤ i ≤ n.Then for 1 ≤ i ≤ n,f(Hi) = f(vi−1) + f(vi) +∑f(V ′i ) +∑f(Ei)= f(vi−1) + f(vi) +∑Xn−i+1 +∑Yn−i+1=n(p+ q)2 + 3(p+ q)− 2n(p+ q) + 2n− 22As Hi ∼= H for 1 ≤ i ≤ n, Hn is H-supermagic.Suppose p is even, q is odd and n is odd. Since p− 2 is even and n is odd, by(c) there exists an n-equipartition P ′1 = {X ′1, X ′2, . . . , X ′n} of [1, n(p− 2)] suchthat ∑P ′1 =(p− 2) [n(p− 2) + 1]2+[−n− 12,n− 12].Adding n+1 to [1, n(p− 2)], we get an n-equipartition P1 = {X1, X2, . . . , Xn}of [n+ 2, np− n+ 1] such that such that∑P1 = (p− 2)(n+ 1) + (p− 2) [n(p− 2) + 1]2 +[−n− 12,n− 12]Since q is odd, as in Case (i) there exists an n-equipartition P2 = {Y1, Y2, . . . , Yn}of (np− n+ 1) + [1, nq] such that∑P2 = q(np− n+ 1) + (q − 1)(nq + n+ 1)2 + [1, n]Define a total labeling f : V ∪ E → {1, 2, 3, . . . , np+ nq − n+ 1} as follows:(i) f(vi) = i+ 1 for 0 ≤ i ≤ n.(ii) f(V ′i ) = Xn−i+1 for 1 ≤ i ≤ n.(iii) f(Ei) = Yn−i+1 for 1 ≤ i ≤ n.Then for 1 ≤ i ≤ n,f(Hi) = f(vi−1) + f(vi) +∑f(V ′i ) +∑f(Ei)= f(vi−1) + f(vi) +∑Xn−i+1 +∑Yn−i+1=n(p+ q)2 + 3(p+ q)− 2n(p+ q) + 2n− 22As Hi ∼= H for 1 ≤ i ≤ n, Hn is H-supermagic.30 P. JEYANTHI AND P. SELVAGOPAL§5. IllustrationsA chain of a 2-connected (5, 7) simple graph H of length 5 is shown in Figure1 and a chain of a 2-connected (6, 9) simple graph H of length 3 is shown inFigure 2.uu uuuu uuuuuuuuuuuu21 12611 1043213 25 20 14 24 19946 27475636374538 3528485544 495434293912 uvQQQQQQQ23181622177326415423151524315408 33305053uFigure 1. p = 5, q = 7, s(f) = 322.7u t u u uuuuuuuuuuu u19 8 5 21 1062292342814121611 3015261331322024337384218362431725 33 392729354140341Figure 2. p = 6, q = 9, s(f) = 317.MAGIC COVERING OF A CHAIN OF 2-CONNECTED GRAPH 31References[1] J.A. Gallian, A Dynamic Survey of Graph labeling, The Electronic Journal ofCombinatorics. 5 (2005).[2] A. Gutierrez and A. Llado,Magic coverings, J. Combin. Math. Combin. Com-put. 55 (2005), 43–56.[3] P. Selvagopal, P. Jeyanthi, On Ck-supermagic graphs, International Journalof Mathematics and Computer Science, 3.1 (2008), 25–30.P. JeyanthiDepartment of Mathematics, Govindammal Aditanar College for womenTiruchendur 628 215, IndiaE-mail : jeyajeyanthi@rediffmail.comP. SelvagopalDepartment of Mathematics, Lord Jegannath College of Engineering and TechnologyPSN Nagar, Ramanathichenputhur,629 402, IndiaE-mail : ps gopaal@yahoo.co.in

Magic covering of chain of an arbitrary 2-connected simple graph

https://tus.repo.nii.ac.jp/?action=repository_action_common_download&item_id=1072&item_no=1&attribute_id=22&file_no=1

Magic covering of chain of an arbitrary 2-connected simple graph

Abstract

Similar works

Full text

Available Versions

Tokyo University of Science Repository for Academic Resources / 東京理科大学学術リポジトリ