Review of Literature

Review of Literature Harmonious and Feliticious Labeling. Corona graphs Cn*K1 are highly symmetrical graphs. They are unicyclic graphs. Cycle-related graphs are of major importance for labeling. Some labelings are already defined for unicyclic graphs. Rosa [1] showed that Cn is graceful if and only if n ≡ 0 or 3 (mod 4). Graham and Sloane [2] proved that Cn is harmonious if and only if n ≡ 1 or 3 (mod 4). Lee, Schmeichel and.Shee [3] give some necessary conditions for a graph to be feliticious. The generalization of harmonious graph is feliticious graph is shown. They show that some families of graph like cycles of order 4k, complete bipartite graphs, generalized Petersen graphs etc. are feliticious while some graphs like cycles of order 4k+2, complete graph Kn when n ≥ 5 are not feliticious. Cordial labeling and Divisor cordial labeling Cahit [4] introduced a new vertex labeling for graphs called cordial labeling. He has proved the results of cordial labeling on trees. The cordial labeling of corona Cn*K1 is not yet undertaken. The 2-equitable graphs were called cordial graphs by Cahit [5]. He has shown that the cycle Cn with n-vertices is 3-equitable iff n ≠ 3 (mod 6). The k-angular cactus, Tk,t, is a connected graph whose all blocks are cycles with k-vertices. Cahit showed that Tk,t is cordial iff k.t ≠ 2 (mod 4). Some special classes of graphs such as dragon, corona, wheel, full binary trees, G*K2,n and G*K3,n are shown to be divisor cordial by Varatharajan at. el. [6] Graceful labeling Christian Barrientos [7] defines difference vertex labelings and shows that any forest whose components are caterpillars is odd-graceful. He also shows that every tree of diameter up to 5 is odd graceful. A caterpillar can also be obtained from Cn*K1, by removing a edge from cycle Cn. In [8] Elumalai and Sethvroman proves that every n cycle (n ≥ 6) with parallel chords is graceful for all n ≥ 6 and every n-cycle with parallel Pk – chords of increasing lengths is graceful for n ≡ 2 (mod 4) with 1≤ k ≤ [ -1 . Andrea Vietri [9] exhibits a graceful labeling for generalized Petersen graph P8t,3 with t ≥ 1. As a consequence for any fixed t, a cyclic edge-decomposition of complete graph K48t+1 into copies of P8t,3 is obtained. Due to its extreme versatility this technique is used for finding new graceful labeling not necessarily involving generalized Petersen graphs. Ramfrez-Alfonsin [10] has showed that all caterpillars are graceful. Further he shows that Replicated Cycles Rx (Cn) i.e. the cycle with n vertices is graceful iff n = 0 or 3 (mod 4). Kathieresan [11] have proved the gracefulness of new class of graphs denoted n-1 by Kn*S2 - He also prove that the graphs consisting of 2m+1 internally disjoint path of length 2r each, connecting two fixed vertices are also graceful. Christian Barrientos [12] studied graceful labeling of some graphs that result of two different constructions. The first construction produces chains graphs. It is shown that chain graph is graceful if any graph in the chain has α-labeling. The second construction is the corona graph Cn*nK1. It is shown that corona graph Cn*nK1 is graceful for every positive integer n ≥ 3 & m ≥ 1. Redl [13] has given two mathematical programming formulations of the graceful labeling problem. He has given new results on the gracefulness of three classes of graphs i.e. generalized Petersen graphs P(n,k), double cones Cn + and product graphs of form Kn*Pn. The study of graceful labeling of graphs with pendant edges is given by Christian Barrientos [14]. In particular graceful labeling for graphs of the form G*nK1 and G + nK1 when G is a graceful graph with order greater than its size is given. He has also shown a graceful labeling of unicyclic graph formed by a cycle with any no. of pendant edges attached. Equitability Minimal k-equitability of C2n*K1 ; k = 2, 2n and associated graphs were dealt with by Manisha Acharya and Bhat Nayak. [15]. They proved that C2n*K1, are minimally k-equitable for k = 2, 2n and that C2n+1*K1 are minimally (2n+1) - equitable. Further they showed minimal 2-equitability of graphs that are obtained by removing any set of rays from a certain part of the corona graphs C2n*K1. Bloom has posed a problem at the Graph theory meeting of the New York Academy of Sciences in Nov 1989. He posed the following question: Is the condition that, k is proper divisor of n sufficient for the cycle Cn to have a minimal k-equitable labeling? Jerzy Wojciechowski [16] gives positive answer to this question. He writes “if k and m are integers greater than 1 then the cycle Cmk is minimally k-equitable”. He considers two cases (i) when k is odd and (ii) when k is even. A shell of size n is obtained by taking a cycle of length n and adding exactly (n- 3) chords which are concurrent at a vertex in the initial cycle. This vertex is called apex of the shell. If one-point unions of these shells are taken then it is called multiple shell. Chitre and Limaye [17] have proved that multiple shells are 5-equitable. Christian Barrientos [18] proves that the corona graph Cn*K1 is equitable. He further shows that corona Cm*nK1 is k-equitable for some values of k, m and n. For every proper divisor k of 2n the corona Cn*K1 is k-equitable is shown in [18]. The corona C3*nK1 is complete 3-equitable graph for every positive value of n. If n is odd then corona C3*nK1 is complete 2-equitable graph. These coronas can be further checked for the parameters k, m and n. k-equitable labeling of complete – bipartite graph and multibipartite graph is shown by David Vickrey [19]. He considers k-equitable labeling and shows that K(m,n) is not k-equitable if m+ n ≤ k ≤ mn . Also if 0 ≤ t ≤ k-m and s = 0 then K(m,n) is k-equitable iff k > mn. He further generalizes that for the bipartite graph K(m,n) ; m,n ≥ 3, k ≥ 3, the only graphs which are k-equitable are K4,4 for k = 3 ; K(3,k-1) k and K(m,n) for k > mn. Some more labelings Lee and Shee [20] introduce Skolem graceful labeling of graphs. They show that a tree is Skolem graceful iff it is graceful. Further they make corollary that every caterpillar is Skolem graceful. Also they make corollary that if T is a Skolem graceful tree then the corona ToK1 is Skolem graceful. Juan and Liu [21] obtains antipodal number for the cycle Cn when n ≡ 1 (mod 4). They consider antipodal labeling for the cycle Cn. Chartrand et al[22] determined value of antipodal number of Cn for n ≡ 2 (mod 4). Antipodal number is the minimum span of an antipodal labeling admitted. G. Juan and Liu studied antipodal labeling for cycles given by Chartrand [22] in which lower bound for an(Cn) [antipodal number of Cn] were shown. They proved that for n ≡ 2 (mod 4) the value of an (Cn) was exact as given in [22]. The bound for case n ≡ 1 (mod n) conjectured to be exact value as well [22]. The class of complete graph Kn is considered by Krishnappa [23] which admits a vertex magic total labeling. The technique of generating magic squares and ideas of graph factorization is used in the construction of graph which admit verten magic total labeling attempts to draw connections between graph factorization and graph labeling. Suresh Kumar [24] show that graph Cn*K1,m is sequential for all m and all odd n. Adidarma Sepang et al [25] have constructed new super edge magic total labeling of special classes of unicyclic graphs. This construction is taken from super edge magic total labeling of odd cycles. The authors have shown that all cycle-like unicylics are super edge magic total graph. Further they prove that all coronas like unicycles are super edge magic total graph. Prime labeling of certain classes of graphs are discussed by Ramya, Rangarajan and Sattanathan [26]. They show that H-graph which is a 3 regular graph satisfy prime labeling. A Gear graph is obtained from wheel graph. They have shown that gear graph satisfy prime labeling. The corona Cn*K1,3 and sunflower graph also satisfy the prime labeling is shown with stepwise algorithm. Lai and Chang [27] studies profile of the corona G^H of two graphs G and H. The exact value of the profile of corona G^H are obtained when G has certain properties, including when G is a caterpillar, a complete graph or a cycle. They have shown that if Cn is a cycle of n vertices and H is a graph on M vertices then P(Cn) = 2n-3 and M (Cn) = 2n-2 Also P(Cn^H) = P(Cn) +n P(H) + M (Cn)m = 2n-3+nP(H) + (2n-2)m. It is shown by Hungund and Akka [28] that the corona of an odd cycle Cn on K2 admits super edge magic-labelings and reverse super edge magic-labeling]. Further they have obtained super edge-magic strength as well as reverse super edge-magic strength of Cn*K2 for all odd n≥3. Solairaju and Muruganantham [29] showed even vertex graceful of path, circuit, star, wheel and some extension friendship graphs and helm graph.

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