A Dynamic Survey of Graph Labeling Joseph A. Gallian Department of Mathematics and Statistics University of Minnesota Duluth Duluth, Minnesota 55812 jgallian@d.umn.edu Submitted: September 1, 1996 Accepted: November 14, 1997 Twelfth edition published: January 31, 2009 Mathematics Subject Classifications: 05C78 Abstract A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index. the electronic journal of combinatorics 16 (2009), #DS6 1

Contents 1 Introduction 5 2 Graceful and Harmonious Labelings 7 2.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Cycle-Related Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Product Related Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Disconnected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Joins of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 Miscellaneous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Table 1: Summary of Graceful Results . . . . . . . . . . . . . . . . . . . . 24 Table 2: Summary of Harmonious Results . . . . . . . . . . . . . . . . . . 28 3 Variations of Graceful Labelings 31 3.1 ��-labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Table 3: Summary of Results on ��-labelings . . . . . . . . . . . . . . . . . 37 3.2 k-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 ��-Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Skolem-Graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5 Odd-Graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6 Graceful-like Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.8 The Friendly Index���Balance Index . . . . . . . . . . . . . . . . . . . . . . 55 3.9 k-equitable Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.10 Hamming-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 Variations of Harmonious Labelings 59 4.1 Sequential and Strongly c-harmonious Labelings . . . . . . . . . . . . . . . 59 4.2 (k, d)-arithmetic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 (k, d)-indexable Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Elegant Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Felicitous Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5 Magic-type Labelings 67 5.1 Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Table 4: Summary of Magic Labelings . . . . . . . . . . . . . . . . . . . . 72 5.2 Edge-magic Total and Super Edge-magic Labelings . . . . . . . . . . . . . 74 Table 5: Summary of Edge-magic Total Labelings . . . . . . . . . . . . . . 82 Table 6: Summary of Super Edge-magic Labelings . . . . . . . . . . . . . . 85 5.3 Vertex-magic Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . . 89 Table 7: Summary of Vertex-magic Total Labelings . . . . . . . . . . . . . 95 the electronic journal of combinatorics 16 (2009), #DS6 2

Table 8: Summary of Super Vertex-magic Total Labelings . . . . . . . . . 97 Table 9: Summary of Totally Magic Labelings . . . . . . . . . . . . . . . . 98 5.4 Magic Labelings of Type (a, b, c) . . . . . . . . . . . . . . . . . . . . . . . . 99 Table 10: Summary of Magic Labelings of Type (a, b, c) . . . . . . . . . . . 101 5.5 Other Types of Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . 102 6 Antimagic-type Labelings 105 6.1 Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Table 11: Summary of Antimagic Labelings . . . . . . . . . . . . . . . . . 109 Table 12: Summary of (a, d)-Edge-Antimagic Vertex Labelings . . . . . . . 110 Table 13: Summary of (a, d)-Antimagic Labelings . . . . . . . . . . . . . . 111 6.2 (a, d)-Antimagic Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . 112 Table 14: Summary of (a, d)-Vertex-Antimagic Total and Super (a, d)- Vertex-Antimagic Total Labelings . . . . . . . . . . . . . . . . . . . 119 Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings . . . . . . . . 120 Table 16: Summary of (a, d)-Super-Edge-Antimagic Total Labelings . . . . 122 6.3 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1) . . . . 123 Table 17: Summary of Face Antimagic Labelings . . . . . . . . . . . . . . . 126 Table 18: Summary of d-antimagic Labelings of Type (1,1,1) . . . . . . . . 127 6.4 Product Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . 128 7 Miscellaneous Labelings 129 7.1 Sum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Table 19: Summary of Sum Graph Labelings . . . . . . . . . . . . . . . . . 135 7.2 Prime and Vertex Prime Labelings . . . . . . . . . . . . . . . . . . . . . . 136 Table 20: Summary of Prime Labelings . . . . . . . . . . . . . . . . . . . . 139 Table 21: Summary of Vertex Prime Labelings . . . . . . . . . . . . . . . . 142 7.3 Edge-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Table 22: Summary of Edge-graceful Labelings . . . . . . . . . . . . . . . . 150 7.4 Radio Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 Line-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.6 Representations of Graphs modulo n . . . . . . . . . . . . . . . . . . . . . 153 7.7 k-sequential Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.8 IC-colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.9 Product Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.10 Prime Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.11 Geometric Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.12 Sequentially Additive Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.13 Strongly Multiplicative Graphs . . . . . . . . . . . . . . . . . . . . . . . . 157 7.14 Mean Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.15 Permutation and Combination Graphs . . . . . . . . . . . . . . . . . . . . 159 7.16 Strongly ?-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.17 Irregular total Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 the electronic journal of combinatorics 16 (2009), #DS6 3

7.18 Sigma Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.19 Set Graceful and Set Sequential Graphs . . . . . . . . . . . . . . . . . . . . 161 7.20 Divisor Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.21 Difference Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 References 162 Index 213 the electronic journal of combinatorics 16 (2009), #DS6 4

1 Introduction Most graph labeling methods trace their origin to one introduced by Rosa [733] in 1967, or one given by Graham and Sloane [380] in 1980. Rosa [733] called a function f a ��- valuation of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, . . . , q} such that, when each edge xy is assigned the label |f(x) - f(y)|, the resulting edge labels are distinct. Golomb [371] subsequently called such labelings graceful and this is now the popular term. Rosa introduced ��-valuations as well as a number of other labelings as tools for decomposing the complete graph into isomorphic subgraphs. In particular, ��-valuations originated as a means of attacking the conjecture of Ringel [723] that K2n+1 can be decomposed into 2n + 1 subgraphs that are all isomorphic to a given tree with n edges. Although an unpublished result of Erd�� os says that most graphs are not graceful (cf. [380]), most graphs that have some sort of regularity of structure are graceful. Sheppard [807] has shown that there are exactly q! gracefully labeled graphs with q edges. Rosa [733] has identified essentially three reasons why a graph fails to be graceful: (1) G has ���too many vertices��� and ���not enough edges,��� (2) G ���has too many edges,��� and (3) G ���has the wrong parity.��� An infinite class of graphs that are not graceful for the second reason is given in [170]. As an example of the third condition Rosa [733] has shown that if every vertex has even degree and the number of edges is congruent to 1 or 2 (mod 4) then the graph is not graceful. In particular, the cycles C4n+1 and C4n+2 are not graceful. Acharya [11] proved that every graph can be embedded as an induced subgraph of a graceful graph and a connected graph can be embedded as an induced subgraph of a graceful connected graph. Acharya, Rao, and Arumugam [26] proved: every triangle- free graph can be embedded as an induced subgraph of a triangle-free graceful graph every planar graph can be embedded as an induced subgraph of a planar graceful graph and every tree can be embedded as an induced subgraph of a graceful tree. These results demonstrate that there is no forbidden subgraph characterization of these particular kinds of graceful graphs. Harmonious graphs naturally arose in the study by Graham and Sloane [380] of mod- ular versions of additive bases problems stemming from error-correcting codes. They defined a graph G with q edges to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x) + f(y) (mod q), the resulting edge labels are distinct. When G is a tree, exactly one label may be used on two vertices. Analogous to the ���parity��� necessity con- dition for graceful graphs, Graham and Sloane proved that if a harmonious graph has an even number of edges q and the degree of every vertex is divisible by 2k then q is divisible by 2k+1. Thus, for example, a book with seven pages (i.e., the cartesian product of the complete bipartite graph K1,7 and a path of length 1) is not harmonious. Liu and Zhang [620] have generalized this condition as follows: if a harmonious graph with q edges has degree sequence d1, d2, . . . , dp then gcd(d1, d2, . . . dp, q) divides q(q - 1)/2. They have also proved that every graph is a subgraph of a harmonious graph. More generally, Sethuraman and Elumalai [779] have shown that any given set of graphs G1, G2, . . . , Gt the electronic journal of combinatorics 16 (2009), #DS6 5

can be embedded in a graceful or harmonious graph. Determining whether a graph has a harmonious labeling was shown to be NP-complete by Auparajita, Dulawat, and Rathore in 2001 (see [518]). Over the past three decades in excess of 1000 papers have spawned a bewildering array of graph labeling methods. Despite the unabated procession of papers, there are few general results on graph labelings. Indeed, the papers focus on particular classes of graphs and methods, and feature ad hoc arguments. In part because many of the papers have appeared in journals not widely available, frequently the same classes of graphs have been done by several authors and in some cases the same terminology is used for different concepts. In this article, we survey what is known about numerous graph labeling methods. The author requests that he be sent preprints and reprints as well as corrections for inclusion in the updated versions of the survey. Earlier surveys, restricted to one or two labeling methods, include [159], [175], [495], [346], and [348]. The book edited by Acharya, Arumugam, and Rosa [16] includes a variety of labeling methods that we do not discuss in this survey. The extension of graceful labelings to directed graphs arose in the characterization of finite neofields by Hsu and Keedwell [445], [446]. The relationship between graceful digraphs and a variety of algebraic structures including cyclic difference sets, sequenceable groups, generalized complete mappings, near-complete mappings, and neofields is discussed in [179] and [180]. The connection between graceful labelings and perfect systems of difference sets is given in [162]. Labeled graphs serve as useful models for a broad range of applications such as: coding theory, x-ray crystallography, radar, astronomy, circuit design, communication network addressing, data base management, and models for constraint programming over finite domains���see [176], [177], [895], [710], [855], [856], and [648] for details. Terms and notation not defined below follow that used in [237] and [346]. the electronic journal of combinatorics 16 (2009), #DS6 6

2 Graceful and Harmonious Labelings 2.1 Trees The Ringel-Kotzig conjecture that all trees are graceful has been the focus of many papers. Kotzig [448] has called the effort to prove it a ���disease.��� Among the trees known to be graceful are: caterpillars [733] (a caterpillar is a tree with the property that the removal of its endpoints leaves a path) trees with at most 4 end-vertices [448], [1009] and [462] trees with diameter at most 5 [1009] and [442] trees with at most 27 vertices [38] symmetrical trees (i.e., a rooted tree in which every level contains vertices of the same degree) [163], [708] rooted trees where the roots have odd degree and the lengths of the paths from the root to the leaves differ by at most one and all the internal vertices have the same parity [219] the graph obtained by identifying the endpoints any number of paths of a fixed length except for the case that the length has the form 4r + 1, r 1 and the number of paths is of the form 4m with m r [753] regular bamboo trees [753] (a rooted tree consisting of branches of equal length the endpoints of which are identified with end points of stars of equal size) and olive trees [698], [2] (a rooted tree consisting of k branches, where the ith branch is a path of length i). Aldred, �� Sir���� a n and �� Sir���� a n [39] have proved that the number of graceful labelings of Pn grows at least as fast as (5/3)n. They mention that this fact has an application to topological graph theory. One such application was provided by Goddyn, Richter, and and �� Sir���� a n [369] who used graceful labelings of paths on 2s + 1 vertices (s ��� 2) to obtain 22s cyclic oriented triangular embeddings of the complete graph on 12s + 7 vertices. The Aldred, �� Sir���� a n and �� Sir���� a n bound was improved by Adamaszek [32] to (2.37)n with the aid of a computer. Cattell [227] has shown that when finding a graceful labeling of a path one has almost complete freedom to choose a particular label i for any given vertex v. In particular, he shows that the only cases of Pn when this cannot be done are when n ��� 3 (mod 4) or n ��� 1 (mod 12), v is in the smaller of the two partite sets of vertices, and i = (n - 1)/2. A spider is a tree that has at most one vertex (called the center) of degree greater than 2. Bahls, Lake, and Wertheim [114] proved that spiders for which the lengths of every path from the center to a leaf differ by at most one are graceful and for spiders for which the lengths of every path from the center to a leaf has the same length and there is an odd number of such paths there is a family of graceful labelings. In [312] and [313] Eshghi and Azimi [312] discuss a programming model for finding graceful labelings of large graphs. The computational results show that the models can easily solve the graceful labeling problems for large graphs. They used this method to verify that all trees with 30, 35, or 40 vertices are graceful. Stanton and Zarnke [873] and Koh, Rogers, and Tan [498], [499], [497] gave methods for combining graceful trees to yield larger graceful trees. Rogers in [731] and Koh, Tan, and Rogers in [496] provide recursive constructions to create graceful trees. Burzio and Ferrarese [206] have shown that the graph obtained from any graceful tree by subdividing every edge is also graceful. Morgan [667] has used Skolem sequences to construct classes of graceful trees. In 1979 Bermond [159] conjectured that lobsters are graceful (a lobster is a tree with the property the electronic journal of combinatorics 16 (2009), #DS6 7

that the removal of the endpoints leaves a caterpillar). Morgan [666] has shown that all lobsters with perfect matchings are graceful. Mishra and Panigrahi [662] and [696] found classes of graceful lobsters of diameter at least five. In [783] Sethuraman and Jesintha [783] explores how one can generate graceful lobsters from a graceful caterpillar while in [787] and [788] (see also Jesin) they show how to generate graceful trees from a graceful star. More special cases of Bermond���s conjecture have been done by Ng [681], by Wang, Jin, Lu, and Zhang [944], and by Abhyanker [1]. Morgan and Rees [668] have used Skolem and Hooked-Skolem sequences to generate classes of graceful lobsters. Whether or not lobsters are harmonious seems to have attracted no attention thus far. Barrientos [138] defines a y-tree as a graph obtained from a path by appending an edge to a vertex of a path adjacent to an end point. He proves that graphs obtained from a y-tree T by replacing every edge ei of T by a copy of K2,ni in such a way that the ends of ei are merged with the two independent vertices of K2,ni after removing the edge ei from T are graceful. Bermond and Sotteau [163] have shown that a rooted tree in which every level contains vertices of the same degree (symmetrical trees) are graceful. Sethuraman and Jesintha [784], [785] and [786] (see also [459]) proved that rooted trees obtained by identifying one of the end vertices adjacent to either of the penultimate vertices of any number of caterpillars having equal diameter at least 3 with the property that all the degrees of internal vertices of all such caterpillars have the same parity are graceful. They also proved that rooted trees obtained by identifying either of the penultimate vertices of any number of caterpillars having equal diameter at least 3 with the property that all the degrees of internal vertices of all such caterpillars have the same parity are graceful. In [784], [785], and [786] (see also [459]) Sethuraman and Jesintha prove that all rooted trees in which every level contains pendant vertices and the degrees of the internal vertices in the same level are equal are graceful. Kanetkar and Sane [469] show that trees formed by identifying one end vertex of each of six or fewer paths whose lengths determine an arithmetic progression are graceful. Chen, L�� u, and Yeh [241] define a firecracker as a graph obtained from the concatenation of stars by linking one leaf from each. They also define a banana tree as a graph obtained by connecting a vertex v to one leaf of each of any number of stars (v is not in any of the stars). They proved that firecrackers are graceful and conjecture that banana trees are graceful. Sethuraman and Jesintha [790] and [789] (see also [459] have shown that all banana trees and extended banana trees (graphs obtained by joining a vertex to one leaf of each of any number of stars by a path of length of at least two) are graceful. Various kinds of bananas trees had been shown to be graceful by Bhat-Nayak and Deshmukh [165], by Murugan and Arumugam [675], [673] and by Vilfred [925]. Despite the efforts of many, the graceful tree conjecture remains open even for trees with maximum degree 3. Aldred and McKay [38] used a computer to show that all trees with at most 26 vertices are harmonious. That caterpillars are harmonious has been shown by Graham and Sloane [380]. In a paper published in 2004 Krishnaa [516] claims to proved that all trees have both graceful and harmonious labelings. However, her proofs were flawed. Using a variant of the Matrix Tree Theorem, Whitty [953] specifies an n �� n matrix the electronic journal of combinatorics 16 (2009), #DS6 8

of indeterminates whose determinant is a multivariate polynomial that enumerates the gracefully labelled (n + 1)-vertex trees. Whitty also gives a bijection between gracefully labelled graphs and rook placements on a chessboard on the M�� obius strip. Cahit extended the notion of gracefulness to directed graphs in [220]. More specialized results about trees are contained in [159], [175], [495], [629], [213], [461], and [734]. 2.2 Cycle-Related Graphs Cycle-related graphs have been the major focus of attention. Rosa [733] showed that the n-cycle Cn is graceful if and only if n ��� 0 or 3 (mod 4) and Graham and Sloane [380] proved that Cn is harmonious if and only if n ��� 1 or 3 (mod 4). Wheels Wn = Cn + K1 are both graceful and harmonious ��� [334], [440] and [380]. As a consequence we have that a subgraph of a graceful (harmonious) graph need not be graceful (harmonious). The n-cone (also called the n-point suspension of Cm) Cm + Kn has been shown to be graceful when m ��� 0 or 3 (mod 12) by Bhat-Nayak and Selvam [171]. When n is even and m is 2, 6 or 10 (mod 12) Cm + Kn violates the parity condition for a graceful graph. Bhat-Nayak and Selvam [171] also prove that the following cones are graceful: C4 + Kn, C5 + K2, C7 + Kn, C9 + K2, C11 + Kn and C19 + Kn. The helm Hn is the graph obtained from a wheel by attaching a pendant edge at each vertex of the n-cycle. Helms have been shown to be graceful [55] and harmonious [367], [616], [617] (see also [620], [772], [608], [271] and [715]). Koh, Rogers, Teo, and Yap, [500] define a web graph as one obtained by joining the pendant points of a helm to form a cycle and then adding a single pendant edge to each vertex of this outer cycle. They asked whether such graphs are graceful. This was proved by Kang, Liang, Gao, and Yang [472]. Yang has extended the notion of a web by iterating the process of adding pendant points and joining them to form a cycle and then adding pendant points to the new cycle. In his notation, W (2, n) is the web graph whereas W (t, n) is the generalized web with t n-cycles. Yang has shown that W (3, n) and W (4, n) are graceful (see [472]), Abhyanker and Bhat-Nayak [3] have done W (5, n) and Abhyanker [1] has done W (t, 5) for 5 ��� t ��� 13. Gnanajothi [367] has shown that webs with odd cycles are harmonious. Seoud and Youssef [772] define a closed helm as the graph obtained from a helm by joining each pendant vertex to form a cycle and a flower as the graph obtained from a helm by joining each pendant vertex to the central vertex of the helm. They prove that closed helms and flowers are harmonious when the cycles are odd. A gear graph is obtained from the wheel by adding a vertex between every pair of adjacent vertices of the cycle. In 1984 Ma and Feng [633] proved all gears are graceful while in a Master���s thesis in 2006 Chen [242] proved all gears are harmonious. Liu [616] has shown that if two or more vertices are inserted between every pair of vertices of the n-cycle of the wheel Wn, the resulting graph is graceful. Liu [614] has also proved that the graph obtain from a gear graph by attaching one or more pendant points to each vertex between the cycle vertices is graceful. Abhyanker [1] has investigated various unicyclic (that is, graphs with exactly one cycle) graphs. He proved that the unicyclic graphs obtained by identifying one vertex of C4 with the root of the olive tree with 2n branches and identifying an adjacent vertex on the electronic journal of combinatorics 16 (2009), #DS6 9

C4 with the end point of the path P2n-2 are graceful. He showed that if one attaches any number of pendent edges to these unicyclic graphs at the vertex of C4 that is adjacent to the root of the olive tree but not adjacent to the end vertex of the attached path, the resulting graphs are graceful. Likewise, Abhyanker proved that the graph obtained by deleting the branch of length 1 from an olive tree with 2n branches and identifying the root of the edge deleted tree with a vertex of a cycle of the form C2n+3 is graceful. He also has a number of results similar to these. Delorme, Maheo, Thuillier, Koh, and Teo [274] and Ma and Feng [632] showed that any cycle with a chord is graceful. This was first conjectured by Bodendiek, Schumacher, and Wegner [182], who proved various special cases. In 1985 Koh and Yap [501] generalized this by defining a cycle with a Pk-chord to be a cycle with the path Pk joining two nonconsecutive vertices of the cycle. They proved that these graphs are graceful when k = 3 and conjectured that all cycles with a Pk-chord are graceful. This was proved for k ��� 4 by Punnim and Pabhapote in 1987 [711]. Chen [247] obtained the same result except for three cases which were then handled by Gao [395]. In 2005, Sethuraman and Elmalai [778] defined a cycle with parallel Pk-chords as a graph obtained from a cycle Cn (n ��� 6) with consecutive vertices v0, v1, . . . , vn-1 by adding a disjoint path Pk, (k ��� 3), between each pair of nonadjacent vertices v1vn-1, v2vn-2, . . . , vivn-i, . . . , v��v�� where �� = bn/2c- 1 and �� = bn/2c + 2 if n is odd or �� = bn/2c + 1 if n is even. They proved that every cycle Cn (n ��� 6) with parallel Pk-chords is graceful for k = 3, 4, 6, 8, and 10 and they conjecture that the cycle Cn with parallel Pk-chords is graceful for all even k. Xu [971] proved that all cycles with a chord are harmonious except for C6 in the case where the distance in C6 between the endpoints of the chord is 2. The gracefulness of cycles with consecutive chords have also been investigated. For 3 ��� p ��� n - r, let Cn(p, r) denote the n-cycle with consecutive vertices v1, v2, . . . , vn to which the r chords v1vp, v1vp+1, . . . , v1vp+r-1 have been added. Koh and Punnin [491] and Koh, Rogers, Teo, and Yap [500] have handled the cases r = 2, 3 and n-3 where n is the length of the cycle. Goh and Lim [370] then proved that all remaining cases are graceful. Moreover, Ma [631] has shown that Cn(p, n - p) is graceful when p ��� 0, 3 (mod 4) and Ma, Liu, and Liu [634] have proved other special cases of these graphs are graceful. Ma also proved that if one adds to the graph Cn(3, n - 3) any number ki of paths of length 2 from the vertex v1 to the vertex vi for i = 2, . . . , n, the resulting graph is graceful. Chen [247] has shown that apart from four exceptional cases, a graph consisting of three independent paths joining two vertices of a cycle is graceful. This generalizes the result that a cycle plus a chord is graceful. Liu [613] has shown that the n-cycle with consecutive vertices v1, v2, . . . , vn to which the chords v1vk and v1vk+2 (2 ��� k ��� n - 3) are adjoined is graceful. In [272] Deb and Limaye use the notation C(n, k) to denote the cycle Cn with k cords sharing a common endpoint called the apex. For certain choices of n and k there is a unique C(n, k) graph and for other choices there is more than one graph possible. They call these shell-type graphs and they call the unique graph C(n, n - 3) a shell. Notice that the shell C(n, n - 3) is the same as the fan Fn-1 = Pn-1 + K1. Deb and Limaye define a multiple shell to be a collection of edge disjoint shells that have their apex in common. A multiple shell is said to be balanced with width w if every shell has order w or every shell the electronic journal of combinatorics 16 (2009), #DS6 10

has order w or w + 1. Deb and Limaye [272] have conjectured that all multiple shells are harmonious, and have shown that the conjecture is true for the balanced double shells and balanced triple shells. Yang, Xu, Xi, and Qiao [988] proved the conjecture is true for balanced quadruple shells. Sethuraman and Dhavamani [775] use H(n, t) to denote the graph obtained from the cycle Cn by adding t consecutive chords incident with a common vertex. If the common vertex is u and v is adjacent to u, then for k ��� 1, n ��� 4 and 1 ��� t ��� n - 3, Sethuraman and Dhavamani denote by G(n, t, k) the graph obtained by taking the union of k copies of H(n, k) with the edge uv identified. They conjecture that every graph G(n, t, k) is graceful. They prove the conjecture for the case that t = n - 3. For i = 1, 2, . . . , n let vi,1, vi,2, . . . , vi,2m be the successive vertices of n copies of C2m. Sekar [753] defines a chain of cycles C2m,n as the graph obtained by identifying vi,m and vi+1,m for i = 1, 2, . . . , n - 1. He proves that C6,2k and C8,n are graceful for all k and all n. Barrientos [141] proved that all C8,n, C12,n, and C6,2k are graceful. Truszczynski �� [918] studied unicyclic graphs and proved several classes of such graphs are graceful. Among these are what he calls dragons. A dragon is formed by joining the end point of a path to a cycle (Koh, et al. [500] call these tadpoles Kim and Park [487] call them kites). This work led Truszczynski �� to conjecture that all unicyclic graphs except Cn, where n ��� 1 or 2 (mod 4), are graceful. Guo [394] has shown that dragons are graceful when the length of the cycle is congruent to 1 or 2 (mod 4). In his Master���s thesis, Doma [283] investigates the gracefulness of various unicyclic graphs where the cycle has up to 9 vertices. Because of the immense diversity of unicyclic graphs, a proof of Truszczynski���s�� conjecture seems out of reach in the near future. Cycles that share a common edge or a vertex have received some attention. Murugan and Arumugan [674] have shown that books with n pentagonal pages (i.e., n copies of C5 with an edge in common) are graceful when n is even and not graceful when n is odd. Let Cnt) ( denote the one-point union of t cycles of length n. Bermond, Brouwer, and Germa [160] and Bermond, Kotzig, and Turgeon [162]) proved that C3t) ( (that is, the friendship graph or Dutch t-windmill) is graceful if and only if t ��� 0 or 1 (mod 4) while Graham and Sloane [380] proved C3t) ( is harmonious if and only if t 6��� 2 (mod 4). Koh, Rogers, Lee, and Toh [492] conjecture that Cnt) ( is graceful if and only if nt ��� 0 or 3 (mod 4). Yang and Lin [980] have proved the conjecture for the case n = 5 and Yang, Xu, Xi, Li and Haque [986] did the case n = 7. Xu, Yang, Li and Xi [973] did the case n = 11. Qian [713] verifies this conjecture for the case that t = 2 and n is even and Yang, Xu, Xi, and Li [987] did the case n = 9. Figueroa-Centeno, Ichishima, and Muntaner-Batle [325] have shown that if m ��� 0 (mod 4) then the one-point union of 2, 3 or 4 copies of Cm admits a special kind of graceful labeling called an ��-labeling (see Section 3.1) and if m ��� 2 (mod 4), then the one-point union of 2 or 4 copies of Cm admits an ��-labeling. Bodendiek, Schumacher, and Wegner [188] proved that the one-point union of any two cycles is graceful when the number of edges is congruent to 0 or 3 modulo 4. (The other cases violate the necessary parity condition.) Shee [803] has proved that C4t) ( is graceful for all t. Seoud and Youssef [770] have shown that the one-point union of a triangle and Cn is harmonious if and only if n ��� 1 (mod 4) and that if the one-point union of two cycles the electronic journal of combinatorics 16 (2009), #DS6 11

is harmonious then the number of edges is divisible by 4. The question of whether this latter condition is sufficient is open. Figueroa-Centeno, Ichishima, and Muntaner-Batle [325] have shown that if G is harmonious then the one-point union of an odd number of copies of G using the vertex labeled 0 as the shared point is harmonious. Sethuraman and Selvaraju [797] have shown that for a variety of choices of points, the one-point union of any number of non-isomorphic complete bipartite graphs is graceful. They raise the question of whether this is true for all choices of the common point. Another class of cycle-related graphs is that of triangular cacti. The block-cutpoint graph of a graph G is a bipartite graph in which one partite set consists of the cut vertices of G, and the other has a vertex bi for each block Bi of G. A block of a graph is a maximal connected subgraph that has no cut-vertex. A triangular cactus is a connected graph all of whose blocks are triangles. A triangular snake is a triangular cactus whose block-cutpoint-graph is a path (a triangular snake is obtained from a path v1, v2, . . . , vn by joining vi and vi+1 to a new vertex wi for i = 1, 2, . . . , n - 1). Rosa [735] conjectured that all triangular cacti with t ��� 0 or 1 (mod 4) blocks are graceful. (The cases where t ��� 2 or 3 (mod 4) fail to be graceful because of the parity condition.) Moulton [669] proved the conjecture for all triangular snakes. A proof of the general case (i.e., all triangular cacti) seems hopelessly difficult. Liu and Zhang [620] gave an incorrect proof that triangular snakes with an odd number of triangles are harmonious whereas triangular snakes with n ��� 2 (mod 4) triangles are not harmonious. Xu [972] subsequently proved that triangular snakes are harmonious if and only if the number of triangles is not congruent to 2 (mod 4). A double triangular snake consists of two triangular snakes that have a common path. That is, a double triangular snake is obtained from a path v1, v2, . . . , vn by joining vi and vi+1 to a new vertex wi for i = 1, 2, . . . , n-1 and to a new vertex ui for i = 1, 2, . . . , n-1. Xi, Yang, and Wang [968] proved that all double triangular snakes are harmonious. Defining an n-polygonal snake analogous to triangular snakes, Sekar [753] has shown that such graphs are graceful when n ��� 0 (mod 4) (n ��� 8) and when n ��� 2 (mod 4) and the number of polygons is even. Gnanajothi [367, pp. 31���34] had earlier shown that quadrilateral snakes are graceful. Grace [378] has proved that K4-snakes are harmonious. Rosa [735] has also considered analogously defined quadrilateral and pentagonal cacti and examined small cases. Barrientos [133] calls a graph a kCn-snake if it is a connected graph with k blocks whose block-cutpoint graph is a path and each of the k blocks is isomorphic to Cn. (When n 3 and k 3 there is more than one kCn-snake.) If a kCn-snake where the path of minimum length that contains all the cut-vertices of the graph has the property that the distance between any two consecutive cut-vertices is bn/2c it is called linear. Barrientos proves that kC4-snakes are graceful and that the linear kC6-snakes are graceful when k is even. He further proves that kC8-snakes and kC12-snakes are graceful in the cases where the distances between consecutive vertices of the path of minimum length that contains all the cut-vertices of the graph are all even and that certain cases of kC4n-snakes and kC5n- snakes are graceful (depending on the distances between consecutive vertices of the path of minimum length that contains all the cut-vertices of the graph). the electronic journal of combinatorics 16 (2009), #DS6 12

Several people have studied cycles with pendant edges attached. Frucht [334] proved that any cycle with a pendant edge attached at each vertex (i.e., a crown) is graceful. Bu, Zhang, and He [204] have shown that any cycle with a fixed number of pendant edges adjoined to each vertex is graceful. Barrientos [137] defines a hairy cycle as a unicyclic graph other than a cycle in which the deletion of any edge of the cycle results in a caterpillar. He proves that all hairy cycles are graceful [137]. This subsumes the result of Bu, Zhang, and He. If G has order n, the corona of G with H, 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. Barrientos [137] also proves: if G is a graceful graph of order m and size m - 1, then G nK1 and G + nK1 are graceful if G is a graceful graph of order p and size q with q p, then (G ��� (q + 1 - p)K1) nK1 is graceful and all unicyclic graphs other than a cycle for which the deletion of any edge from the cycle results in a caterpillar are graceful. In [135] Barrientos proved that helms (graphs obtained from a wheel by attaching one pendant edge to each vertex) are graceful. Grace [377] showed that an odd cycle with one or more pendant edges at each vertex is harmonious and conjectured that C2n K1, an even cycle with one pendant edge attached at each vertex, is harmonious. This conjecture has been proved by Liu and Zhang [619], Liu [616] and [617], Hegde [416], Huang [447], and Bu [195]. Sekar [753] has shown that the graph Cm Pn obtained by attaching the path Pn to each vertex of Cm e is graceful. For any n ��� 3 and any t with 1 ��� t ��� n, let Cn +t denote the class of graphs formed by adding a single pendant edge to t vertices of a cycle of length n. Ropp [732] proved that for every n and t the class Cn +t contains a graceful graph. Gallian and Ropp [346] conjectured that for all n and t, all members of Cn +t are graceful. This was proved by Qian [713] and by Kang, Liang, Gao, and Yang [472]. Of course, this is just a special case of the aforementioned conjecture of Truszczynski �� that all unicyclic graphs except Cn for n ��� 1 or 2 (mod 4) are graceful. Sekar [753] proved that the graph obtained by identifying an endpoint of a star with a vertex of a cycle is graceful. 2.3 Product Related Graphs Graphs that are cartesian products and related graphs have been the subject of many papers. That planar grids, Pm �� Pn, are graceful was proved by Acharya and Gill [20] in 1978 although the much simpler labeling scheme given by Maheo [640] in 1980 for Pm ��P2 readily extends to all grids. In 1980 Graham and Sloane [380] proved ladders, Pm �� P2, are harmonious when m 2 and in 1992 Jungreis and Reid [468] showed that the grids Pm��Pn are harmonious when (m, n) = (2, 2). A few people have looked at graphs obtained from planar grids in various ways. Kathiresan [475] has shown that graphs obtained from ladders by subdividing each step exactly once are graceful and that graphs obtained by appending an edge to each vertex of a ladder are graceful [477]. Acharya [14] has shown that certain subgraphs of grid graphs are graceful. Lee [531] defines a Mongolian tent as a graph obtained from Pm �� Pn, n odd, by adding one extra vertex above the grid and the electronic journal of combinatorics 16 (2009), #DS6 13

joining every other vertex of the top row of Pm �� Pn to the new vertex. A Mongolian village is a graph formed by successively amalgamating copies of Mongolian tents with the same number of rows so that adjacent tents share a column. Lee proves that Mongolian tents and villages are graceful. A Young tableau is a subgraph of Pm �� Pn obtained by retaining the first two rows of Pm �� Pn and deleting vertices from the right hand end of other rows in such a way that the lengths of the successive rows form a nonincreasing sequence. Lee and Ng [547] have proved that all Young tableaus are graceful. Lee [531] has also defined a variation of Mongolian tents by adding an extra vertex above the top row of a Young tableau and joining every other vertex of that row to the extra vertex. He proves these graphs are graceful. Prisms are graphs of the form Cm �� Pn. These can be viewed as grids on cylinders. In 1977 Bodendiek, Schumacher, and Wegner [182] proved that Cm �� P2 is graceful when m ��� 0 (mod 4). According to the survey by Bermond [159], Gangopadhyay and Rao Hebbare did the case that m is even about the same time. In a 1979 paper, Frucht [334] stated without proof that he had done all Cm ��P2. A complete proof of all cases and some related results were given by Frucht and Gallian [337] in 1988. In 1992 Jungreis and Reid [468] proved that all Cm �� Pn are graceful when m and n are even or when m ��� 0 (mod 4). Yang and Wang have shown that the prisms C4n+2 �� P4m+3 [985], Cn �� P2 [983], and C6 �� Pm(m ��� 2) (see [985]) are graceful. Singh [833] proved that C3 �� Pn is graceful for all n. In their 1980 paper Graham and Sloane [380] proved that Cm �� Pn is harmonious when n is odd and they used a computer to show C4 ��P2, the cube, is not harmonious. In 1992 Gallian, Prout, and Winters [350] proved that Cm �� P2 is harmonious when m = 4. In 1992, Jungreis and Reid [468] showed that C4 �� Pn is harmonious when n ��� 3. Huang and Skiena [449] have shown that Cm �� Pn is graceful for all n when m is even and for all n with 3 ��� n ��� 12 when m is odd. Abhyanker [1] proved that the graphs obtained from C2m+1 �� P5 by adding a pendent edge to each vextex of an outercycle is graceful. Torus grids are graphs of the form Cm �� Cn (m 2, n 2). Very little success has been achieved with these graphs. The graceful parity condition is violated for Cm �� Cn when m and n are odd and the harmonious parity condition [380, Theorem 11] is violated for Cm �� Cn when m ��� 1, 2, 3 (mod 4) and n is odd. In 1992 Jungreis and Reid [468] showed that Cm �� Cn is graceful when m ��� 0 (mod 4) and n is even. A complete solution to both the graceful and harmonious torus grid problems will most likely involve a large number of cases. There has been some work done on prism-related graphs. Gallian, Prout, and Winters [350] proved that all prisms Cm ��P2 with a single vertex deleted or single edge deleted are graceful and harmonious. The M�� obius ladder Mn is the graph obtained from the ladder Pn �� P2 by joining the opposite end points of the two copies of Pn. In 1989 Gallian [345] showed that all M�� obius ladders are graceful and all but M3 are harmonious. Ropp [732] has examined two classes of prisms with pendant edges attached. He proved that all Cm �� P2 with a single pendant edge at each vertex are graceful and all Cm �� P2 with a single pendant edge at each vertex of one of the cycles are graceful. Another class of cartesian products that has been studied is that of books and ���stacked��� books. The book Bm is the graph Sm �� P2 where Sm is the star with m + 1 vertices. In the electronic journal of combinatorics 16 (2009), #DS6 14

1980 Maheo [640] proved that the books of the form B2m are graceful and conjectured that the books B4m+1 were also graceful. (The books B4m+3 do not satisfy the graceful parity condition.) This conjecture was verified by Delorme [273] in 1980. Maheo [640] also proved that Ln ��P2 and B2m ��P2 are graceful. Both Grace [376] and Reid (see [349]) have given harmonious labelings for B2m. The books B4m+3 do not satisfy the harmo- nious parity condition [380, Theorem 11]. Gallian and Jungreis [349] conjectured that the books B4m+1 are harmonious. Gnanajothi [367] has verified this conjecture by showing B4m+1 has an even stronger form of labeling ��� see Section 4.1. Liang [596] also proved the conjecture. In 1988 Gallian and Jungreis [349] defined a stacked book as a graph of the form Sm �� Pn. They proved that the stacked books of the form S2m �� Pn are graceful and posed the case S2m+1 �� Pn as an open question. The n-cube K2 �� K2 ���� �� ���� K2 (n copies) was shown to be graceful by Kotzig [507]���see also [640]. In 1986 Reid [722] found a har- monious labeling for K4 �� Pn. Petrie and Smith [700] have investigated graceful labelings of graphs as an exercise in constraint programming satisfaction. They have shown that Km �� Pn is graceful for (m, n) = (4, 2), (4, 3), (4, 4), (4, 5), and (5,2) but is not graceful for (3, 3) and (6, 2). Their labeling for K5 �� P2 is the unique graceful labeling. They also considered the graph obtained by identifying the hubs of two copies of Wn. The resulting graph is not graceful when n = 3 but is graceful when n is 4 and 5. Smith [855] has used a computer search to prove that Km �� P2 is not graceful for m = 7, 8, 9, and 10. She conjectures that Km �� P2 is not graceful for m 5. For a bipartite graph G with partite sets X and Y let G0 be a copy of G and X0 and Y 0 be copies of X and Y . Lee and Liu [543] define the mirror graph, M(G), of G as the disjoint union of G and G0 with additional edges joining each vertex of Y to its corresponding vertex in Y 0. The case that G = Km,n is more simply denoted by M(m, n). They proved that for many cases M(m, n) has a stronger form of graceful labeling (see ��3.1 for details). The composition G1[G2] is the graph having vertex set V (G1) �� V (G2) and edge set {(x1, y1), (x2, y2)| x1x2 ��� E(G1) or x1 = x2 and y1y2 ��� E(G2)}. The symmetric product G1 ��� G2 of graphs G1 and G2 is the graph with vertex set V (G1) �� V (G2) and edge set {(x1, y1), (x2, y2)| x1x2 ��� E(G1) or y1y2 ��� E(G2) but not both}. Seoud and Youssef [771] have proved that Pn ��� K2 is graceful when n 1 and Pn[P2] is harmonious for all n. They also observe that the graphs Cm ��� Cn and Cm[Cn] violate the parity conditions for graceful and harmonious graphs when m and n are odd. 2.4 Complete Graphs The questions of the gracefulness and harmoniousness of the complete graphs Kn have been answered. In each case the answer is positive if and only if n ��� 4 ([371], [832], [380], [164]). Both Rosa [733] and Golomb [371] proved that the complete bipartite graphs Km,n are graceful while Graham and Sloane [380] showed they are harmonious if and only if m or n = 1. Aravamudhan and Murugan [51] have shown that the complete tripartite graph K1,m,n is both graceful and harmonious while Gnanajothi [367, pp. 25���31] has shown that K1,1,m,n is both graceful and harmonious and K2,m,n is graceful. Some of the same results the electronic journal of combinatorics 16 (2009), #DS6 15