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 Fifteenth edition, December 7, 2012 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 1500 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 (2010), #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 . 14 2.4 Complete Graphs . 16 2.5 Disconnected Graphs . 18 2.6 Joins of Graphs . 20 2.7 Miscellaneous Results . 21 2.8 Summary . 24 Table 1: Summary of Graceful Results . 26 Table 2: Summary of Harmonious Results . 30 3 Variations of Graceful Labelings 33 3.1 α-labelings . 33 Table 3: Summary of Results on α-labelings . 39 3.2 k-graceful Labelings . 40 3.3 γ-Labelings . 42 3.4 Skolem-Graceful Labelings . 42 3.5 Odd-Graceful Labelings . 44 3.6 Graceful-like Labelings . 46 3.7 Cordial Labelings . 51 3.8 The Friendly Index{Balance Index . 59 3.9 Hamming-graceful Labelings . 65 4 Variations of Harmonious Labelings 65 4.1 Sequential and Strongly c-harmonious Labelings . 65 4.2 (k; d)-arithmetic Labelings . 68 4.3 (k; d)-indexable Labelings . 69 4.4 Elegant Labelings . 71 4.5 Felicitous Labelings . 72 4.6 Odd Harmonious and Even Harmonious Labelings . 74 5 Magic-type Labelings 75 5.1 Magic Labelings . 75 Table 4: Summary of Magic Labelings . 80 5.2 Edge-magic Total and Super Edge-magic Total Labelings . 82 Table 5: Summary of Edge-magic Total Labelings . 94 Table 6: Summary of Super Edge-magic Labelings . 97 5.3 Vertex-magic Total Labelings . 101 Table 7: Summary of Vertex-magic Total Labelings . 107 Table 8: Summary of Super Vertex-magic Total Labelings . 109 Table 9: Summary of Totally Magic Labelings . 109 5.4 Magic Labelings of Type (a; b; c) ...........................110 the electronic journal of combinatorics 16 (2010), #DS6 2 Table 10: Summary of Magic Labelings of Type (a; b; c) . 111 5.5 Sigma Labelings/1-vertex magic labelings . 112 5.6 Other Types of Magic Labelings . 112 6 Antimagic-type Labelings 119 6.1 Antimagic Labelings . 119 Table 11: Summary of Antimagic Labelings . 124 Table 12: Summary of (a; d)-Edge-Antimagic Vertex Labelings . 125 Table 13: Summary of (a; d)-Antimagic Labelings . 126 6.2 (a; d)-Antimagic Total Labelings . 127 Table 14: Summary of (a; d)-Vertex-Antimagic Total and Super (a; d)-Vertex- Antimagic Total Labelings . 135 Table 15: Summary of (a; d)-Edge-Antimagic Total Labelings . 136 Table 16: Summary of (a; d)-Super-Edge-Antimagic Total Labelings . 138 6.3 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1) . 139 Table 17: Summary of Face Antimagic Labelings . 141 Table 18: Summary of d-antimagic Labelings of Type (1,1,1) . 142 6.4 Product Antimagic Labelings . 143 7 Miscellaneous Labelings 144 7.1 Sum Graphs . 144 Table 19: Summary of Sum Graph Labelings . 150 7.2 Prime and Vertex Prime Labelings . 151 Table 20: Summary of Prime Labelings . 155 Table 21: Summary of Vertex Prime Labelings . 158 7.3 Edge-graceful Labelings . 159 Table 22: Summary of Edge-graceful Labelings . 166 7.4 Radio Labelings . 167 7.5 Line-graceful Labelings . 168 7.6 Representations of Graphs modulo n . 169 7.7 k-sequential Labelings . 170 7.8 IC-colorings . 170 7.9 Product Cordial Labelings . 171 7.10 Prime Cordial Labelings . 174 7.11 Geometric Labelings . 175 7.12 Strongly Multiplicative Graphs . 176 7.13 Mean Labelings . 177 7.14 Irregular Total Labelings . 180 7.15 Minimal k-rankings . 182 7.16 Set Graceful and Set Sequential Graphs . 183 7.17 Vertex Equitable Graphs . 184 7.18 Sequentially Additive Graphs . 184 7.19 Difference Graphs . 185 7.20 Square Sum Labelings . 185 7.21 Permutation and Combination Graphs . 185 7.22 Strongly *-graphs . 186 the electronic journal of combinatorics 16 (2010), #DS6 3 7.23 Triangular Sum Graphs . 186 7.24 Divisor Graphs . 186 References 186 Index 268 the electronic journal of combinatorics 16 (2010), #DS6 4 1 Introduction Most graph labeling methods trace their origin to one introduced by Rosa [1033] in 1967, or one given by Graham and Sloane [503] in 1980. Rosa [1033] 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 f0; 1; : : : ; qg such that, when each edge xy is assigned the label jf(x) − f(y)j, the resulting edge labels are distinct. Golomb [494] 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 [1022] 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 (see [503]), most graphs that have some sort of regularity of structure are graceful. Sheppard [1130] has shown that there are exactly q! gracefully labeled graphs with q edges. Rosa [1033] 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." The disjoint union of trees is a case where there are too many vertices for the number of edges. An infinite class of graphs that are not graceful for the second reason is given in [236]. As an example of the third condition Rosa [1033] 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 [503] 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. They proved that almost all graphs are not harmonious. 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 [854] 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 [1100] 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 [715]). In the early 1980s Bloom and Hsu [245], [246],[230] 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 f0; 1; 2;:::; jEjg such that θ(y) − θ(x) mod (jEj + 1) is distinct for every edge xy in E. the electronic journal of combinatorics 16 (2010), #DS6 5 Graceful labelings of directed graphs also arose in the characterization of finite neofields by.