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A Dynamic Survey of Graph Labeling A Dynamic Survey of Graph Labeling Joseph A. Gallian Department of Mathematics and Statistics University of Minnesota Duluth Duluth, Minnesota 55812 [email protected] Submitted: September 1, 1996; Accepted: November 14, 1997 Fourteenth edition, November 17, 2011 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 18 (2011), #DS6 1 Contents 1 Introduction 5 2 Graceful and Harmonious Labelings 7 2.1 Trees........................................... 7 2.2 Cycle-RelatedGraphs . .. .. .. .. .. .. .. .. .... 9 2.3 ProductRelatedGraphs. 14 2.4 CompleteGraphs.................................. 16 2.5 DisconnectedGraphs.............................. 17 2.6 JoinsofGraphs................................... 19 2.7 MiscellaneousResults . .. .. .. .. .. .. .. ..... 20 2.8 Summary ........................................ 24 Table1: SummaryofGracefulResults . 25 Table2: SummaryofHarmoniousResults . 29 3 Variations of Graceful Labelings 32 3.1 α-labelings ....................................... 32 Table 3: Summary of Results on α-labelings..................... 38 3.2 k-gracefulLabelings ................................. 39 3.3 γ-Labelings ....................................... 41 3.4 Skolem-GracefulLabelings. ....... 41 3.5 Odd-GracefulLabelings . ..... 43 3.6 Graceful-likeLabelings. ....... 44 3.7 CordialLabelings................................ 49 3.8 TheFriendlyIndex–BalanceIndex . ....... 56 3.9 Hamming-gracefulLabelings. ....... 61 4 Variations of Harmonious Labelings 61 4.1 Sequential and Strongly c-harmoniousLabelings . 61 4.2 (k, d)-arithmeticLabelings. 64 4.3 (k, d)-indexableLabelings . .. .. .. .. .. .. .. .. .. 65 4.4 ElegantLabelings................................ 66 4.5 FelicitousLabelings. ...... 68 4.6 Odd-harmoniousLabelings . ..... 70 5 Magic-type Labelings 70 5.1 MagicLabelings .................................. 70 Table4: SummaryofMagicLabelings . 75 5.2 Edge-magic Total and Super Edge-magic Total Labelings . ............ 77 Table 5: Summary of Edge-magic Total Labelings . ....... 89 Table 6: Summary of Super Edge-magic Labelings . ...... 92 5.3 Vertex-magic Total Labelings . ....... 96 Table 7: Summary of Vertex-magic Total Labelings . ........ 102 Table 8: Summary of Super Vertex-magic Total Labelings . ......... 104 Table 9: Summary of Totally Magic Labelings . 104 5.4 Magic Labelings of Type (a, b, c) ...........................105 the electronic journal of combinatorics 18 (2011), #DS6 2 Table 10: Summary of Magic Labelings of Type (a, b, c) .............. 107 5.5 Sigma Labelings/1-vertex magic labelings . ........... 108 5.6 OtherTypesofMagicLabelings . 108 6 Antimagic-type Labelings 115 6.1 AntimagicLabelings .............................. 115 Table 11: Summary of Antimagic Labelings . 120 Table 12: Summary of (a, d)-Edge-Antimagic Vertex Labelings . 121 Table 13: Summary of (a, d)-Antimagic Labelings . 122 6.2 (a, d)-Antimagic Total Labelings . 123 Table 14: Summary of (a, d)-Vertex-Antimagic Total and Super (a, d)-Vertex- AntimagicTotalLabelings. 131 Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings . 132 Table 16: Summary of (a, d)-Super-Edge-Antimagic Total Labelings . 134 6.3 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1) . 135 Table 17: Summary of Face Antimagic Labelings . 137 Table 18: Summary of d-antimagic Labelings of Type (1,1,1) . 138 6.4 ProductAntimagicLabelings . 139 7 Miscellaneous Labelings 140 7.1 SumGraphs....................................... 140 Table19: SummaryofSumGraphLabelings . 146 7.2 PrimeandVertexPrimeLabelings . 147 Table20: SummaryofPrimeLabelings . 150 Table 21: Summary of Vertex Prime Labelings . 153 7.3 Edge-gracefulLabelings . 154 Table 22: Summary of Edge-graceful Labelings . ....... 160 7.4 RadioLabelings.................................. 161 7.5 Line-gracefulLabelings. ....... 162 7.6 Representations of Graphs modulo n ......................... 163 7.7 k-sequentialLabelings . .. .. .. .. .. .. .. .. 164 7.8 IC-colorings.................................... 164 7.9 ProductCordialLabelings. 165 7.10 PrimeCordialLabelings . 166 7.11 GeometricLabelings . 166 7.12 Sequentially Additive Graphs . ........ 167 7.13 Strongly Multiplicative Graphs . ......... 167 7.14MeanLabelings.................................. 168 7.15 IrregularTotalLabelings. ........ 170 7.16 Minimal k-rankings................................... 172 7.17 Set Graceful and Set Sequential Graphs . ......... 173 7.18 DifferenceGraphs ................................ 174 7.19 SquareSumLabelings . .. .. .. .. .. .. .. .. 174 7.20 Permutation and Combination Graphs . ........ 174 7.21 Strongly*-graphs. .. .. .. .. .. .. .. .. 175 7.22 TriangularSumGraphs . 175 the electronic journal of combinatorics 18 (2011), #DS6 3 7.23DivisorGraphs .................................. 175 References 176 Index 249 the electronic journal of combinatorics 18 (2011), #DS6 4 1 Introduction Most graph labeling methods trace their origin to one introduced by Rosa [973] in 1967, or one given by Graham and Sloane [484] in 1980. Rosa [973] called a function f a β-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 [475] subsequently called such labelings graceful and this is now the popular term. Rosa introduced β-valuations as well as a number of other labelings as tools for decomposing the complete graph into isomorphic subgraphs. In particular, β-valuations originated as a means of attacking the conjecture of Ringel [962] 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. [484]), most graphs that have some sort of regularity of structure are graceful. Sheppard [1059] has shown that there are exactly q! gracefully labeled graphs with q edges. Rosa [973] 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 [229]. As an example of the third condition Rosa [973] 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 [12] 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 [30] 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 [484] of modular 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 condition 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 [812] have generalized this condition as follows: if a harmonious graph with q edges has degree sequence d , d ,...,d then gcd(d , d ,...d ,q) divides q(q 1)/2. 1 2 p 1 2 p − They have also proved that every graph is a subgraph of a harmonious graph. More generally, Sethuraman and Elumalai [1029] have shown that any given set of graphs G1, G2,...,Gt 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 [683]). In the early 1980s Bloom and Hsu [238], [239],[223] extended graceful labelings to directed graphs by defining a graceful labeling on a directed graph D(V, E) as a one-to-one map θ from V to 0, 1, 2,..., E such that θ(y) θ(x) mod ( E + 1) is distinct for every edge xy in E. { | |} − | | Graceful labelings of directed graphs also arose in the characterization of finite
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