Academia.eduAcademia.edu

Outline

Title
Abstract
References
All Topics
Mathematics
Combinatorics

On Additive Bases and Harmonious Graphs

Profile image of Neil SloaneNeil Sloane

SIAM Journal on Algebraic Discrete Methods

https://doi.org/10.1137/0601045
visibility

…

description

23 pages

link

1 file

Abstract

This paper first considers several types of additive bases. A typical problem is to find nv(k), the largest n for which there exists a set {0 al < a2 <" < ak} Of distinct integers modulo n such that each in the range 0 =<-< n can be written at least once as mai + aj (modulo n) with </'. For example, nv(8) 24, as illustrated by the set {0, 1, 2, 4, 8, 13, 18, 22}. The other problems arise if at least is changed to at most, or </' to-</', or if the words modulo n are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge sums are also distinct (modulo n). Some infinite families of graphs (odd cycles, ladders, wheels,...) are shown to be harmonious while others (even cycles, most complete or complete bipartite graphs, .) are not. In fact most graphs are not harmonious. The function nv(k) is the size of the largest harmonious subgraph of the complete graph on k vertices.

Related papers

On the Sumset Labeling of Graphs
Sudev Naduvath

For a positive integer $n$, let $\mZ$ be the set of all non-negative integers modulo $n$ and $\sP(\mZ)$ be its power set. A sumset valuation or a sumset labeling of a given graph $G$ is an injective function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ defined by $f^+ (uv) = f(u)+ f(v)$. A sumset indexer of a graph $G$ is an injective sumset valued function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ is also injective. In this paper, some properties and characteristics of this type of sumset labeling of graphs are being studied.

View PDFchevron_right
On Certain Arithmetic Integer Additive Set-Indexers of Graphs
K. A. Germina

Discrete Mathematics, Algorithms and Applications, 2015

Let ℕ0 denote the set of all non-negative integers and 𝒫(ℕ0) be its power set. An integer additive set-indexer (IASI) of a graph G is an injective function f : V(G) → 𝒫(ℕ0) such that the induced function f+ : E(G) → 𝒫(ℕ0) defined by f+ (uv) = f(u) + f(v) is also injective. A graph G which admits an IASI is called an integer additive set-indexed graph (IASI-graph). An IASI of a graph G is said to be an arithmetic IASI if the elements of the set-labels of all vertices and edges of G are in arithmetic progressions. In this paper, we discuss about two special types of arithmetic IASIs.

View PDFchevron_right
On Integer Additive set-Sequential Graphs
K. A. Germina

arXiv: Combinatorics, 2014

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set of non-negative integers and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a set-sequential labeling of $G$ if $f^{\oplus}(V(G)\cup E(G))=\mathcal{P}(X)-\{\emptyset\}$. A graph $G$ which admits a set-sequential labeling is called a set-sequential graph. An integer additive set-labeling is an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$, $\mathbb{N}_0$ is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set...

View PDFchevron_right
Modular Sumset Labelling of Graphs
Sudev Naduvath

2020

Graph labelling is an assignment of labels or weights to the vertices and/or edges of a graph. For a ground set X of integers, a sumset labelling of a graph is an injective map f:VG→PX such that the induced function f⊕:EG→PX is defined by f+uv=fu+fv, for all uv∈EG, where fu+fv is the sumset of the set-label, the vertices u and v. In this chapter, we discuss a special type of sumset labelling of a graph, called modular sumset labelling and its variations. We also discuss some interesting characteristics and structural properties of the graphs which admit these new types of graph labellings.

View PDFchevron_right
A review on graceful and sequential integer additive set-labeled graphs
K. A. Germina

Cogent Mathematics, 2016

Let ℕ 0 be the set of all non-negative integers and be its power set. Then, an integer additive set-labeling (IASL) of a graph G is an injective function f :V(G) → (ℕ 0), such that the induced function f + :E(G) → (ℕ 0) defined by f + (uv) = f (u) + f (v) and an integer additive set-indexer (IASI) is an integer additive set-labeling such that the induced function f + :E(G) → (ℕ 0) is also injective. An integer additive set-labeling (or an integer additive set-indexer) is said to be a graceful integer additive set-labeling (or graceful integer additive set-indexer) if f + (E(G)) = (X) − {�, {0}} and an integer additive set-labeling (or an integer additive set-indexer) is said to be a sequential integer additive set-labeling (or sequential integer additive set-indexer) if f (V(G)) ∪ f + (E(G)) = (X) − {�}. In this write-up, we creatively and critically review certain studies made on graceful integer additive set-labeling and sequential integer additive set-labeling of given graphs and certain properties and characteristics of the graphs which satisfy this type of set-labelings.

View PDFchevron_right
Some New Results on Weak Integer Additive Set-Labeling of Graphs
Germina Augusthy

Let N 0 denote the set of all non-negative integers and P(N 0) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective function f : V (G) → P(N 0) such that the induced function f + : E(G) → P(N 0) is defined by f + (uv) = f (u)+f (v), where f (u)+f (v) is the sumset of f (u) and f (v). An IASL f is said to be an integer additive set-indexer (IASI) if the associated edge-function f + is also injective. An IASL f of a given graph G is said to be a weak integer additive set-labeling (WIASL) of G if the cardinality of the set-label of every edge of G is equal to the cardinality of the set-label of at least one end vertex of it. In this paper, we study the admissibility of weak integer additive set-labeling by different graphs.

View PDFchevron_right
The domatic number problem on some perfect graph families
Ron Shamir

Information Processing Letters, 1994

An extremely simple, linear time algorithm is given for constructing a domatic partition in totally balanced hypergraphs. This simplies and generalizes previous algorithms for interval and strongly chordal graphs. On the other hand, the domatic number problem is shown to be NP-complete for several families of perfect graphs, including chordal and bipartite graphs.

View PDFchevron_right
On Integer Additive Set-Filtered Graphs
Sudev Naduvath, K. A. Germina

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. In this paper, we introduce the notion of a particular type of integer additive set-indexers called integer additive set-filtered labeling of given graphs and study their characteristics.

View PDFchevron_right
Arithmetic Integer Additive Set-Indexers of Graph Operations
K. A. Germina

Journal of Advanced Research in Pure Mathematics

Let N 0 be the set of non-negative integers. An integer additive set-indexer (IASI) of a graph G is an injective function f : V (G) → 2 N0 such that the induced function f + : E(G) → 2 N0 defined by f + (uv) = f (u) + f (v); uv ∈ E(G), is also injective, where f (u) + f (v) is the sum set of the sets f (u) and f (v). A graph G which admits an IASI is called an IASI graph. An arithmetic IASI is an IASI f , under which the elements of the set-labels of all vertices and edges of a given graph G are in arithmetic progressions. In this paper, we discuss about admissibility of arithmetic IASIs by certain operations and products of graphs.

View PDFchevron_right
On the harmonious chromatic number of a graph
John Mitchem

Discrete Mathematics, 1989

The harmonious chromatic number of a graph G, denoted by h(G), is the least number of colon which can be assigned to the vertices of G such that adjacent vertices are colored differently and any two distinct edges have different color pairs. This is a slight variation of a definition given independently by Hopcroft and Krishnamoorthy and by Frank, Harary, and Plantholt. D. Johnson has shown that determining h(G) is an NP-complete problem. In this paper we give various other theorems on harmonious chromatic number and discuss various open questions.

View PDFchevron_right

References (87)

  1. R. ALTER AND J. A. BARNETT, Remarks on the postage stamp problem with applications to computers, in Proc. 8th Southeastern Conference on Combinatorics, Graph Theory, Computing, Congress Num. XIX, Utilitas Math. Pub., Winnipeg, 1977, pp. 43-59.
  2. A postage stamp problem, Amer. Math. Monthly, 87 (1980), pp. 206-209.
  3. G. BARON, ber Verallgemeinerungen des Langford'schen Problems, in Combinatorial Theory /nd its Applications, P. Erd6s et al., eds., Colloq. Math. Soc. Jinos Bolyai, 4, North-Holland, Amsterdam, 3 vols., 1970, pp. 81-92.
  4. L. D. BAUMERT, Cyclic Difference Sets, Lecture Notes in Math., 182, Springer-Verlag, Berlin, 1971.
  5. J.-C. BERMOND, Problem, in Combinatorics, A. Hajnal and V. T. S6s, eds., Colloq. Math. Soc. Jnos Bolyai, 18, North-Holland, Amsterdam, 2 vols., 1978, p. 1190.
  6. J.-C. BERMOND, A. E. BROUWER AND A. GERMA, Systkmes de triplets et diffr.ences associes, Probl6mes Combinatoires et Th6orie des Graphs, Colloq. Intern. du Centre National de la Rech. Scient., 260, Editions du Centre Nationale de la Recherche Scientifique, Paris, 1978, pp. 35-38.
  7. J.-C. BERMOND, m. KOTZIG AND J. TURGEON, On a combinatorial problem of antennas in radioastronomy, in Combinatorics, A. Hajnal and V. T. S6s, eds., Colloq. Math. Soc. Jinos Bolyai, 18, North-Holland, Amsterdam, 2 vols., 1978, pp. 135-149.
  8. M. R. BEST, A. E. BROUWER, F. J. MACWILLIAMS, A. M. ODLYZKO AND N. J. A. SLOANE, Bounds for binary codes of length less than 25, IEEE Trans. Information Theory, IT-24 (1978), pp. 81-93.
  9. G. S. BLOOM, A chronology of the Ringet-Kotzig conjecture and the continuing quest to call all trees graceful, Ann. N.Y. Acad. Sci., 326 (1979), pp. 32-51.
  10. G. S. BLOOM AND S. W. GOLOMB, Numbered complete graphs, unusual rulers, and assorted applications, in Theory and Applications of Graphs, Lecture Notes in Math., 642, Springer- Verlag, New York, 1978, pp. 53-65.
  11. Applications of numbered undirected graphs, Proc. IEEE, 65 (1977), pp. 562-570.
  12. R. BODENDIEK, H. SCHUMACHER AND H. WEGNER, ber grazib'se Graphen, Math.-Phys. Semes- terberichte, 24 (1977), pp. 103-126.
  13. R. C. BOSE AND S. CHOWLA, Theorems in the additive theory of numbers, Comment. Math. Helvet., 37 (1962-63), pp. 141-147.
  14. N. G DE BRUIJN, On the factorization offinite Abelian groups, Proc. Kon. Ned. Akad. Wetensch. Amsterdam, 56A (1953), pp. 258-264.
  15. On the factorization of cyclic groups, ibid., pp. 370-377.
  16. On number systems, Nieuw Archief voor Wiskunde, (3), IV (1956), pp. 15-17.
  17. J. CAHIT AND R. CAHIT, On the graceful numbering ofspanning trees, Information Processing Letters, 3 (1975), pp. 115-118.
  18. L. CARLITZ AND L. MOSER, On some specialfactorizations of (1 -x")/(1--x), Cana'd. Math. Bull. 9 (1966), pp. 421-426.
  19. R. O. DAVIES, On Langford's problem (//), Math. Gazette, 43 (1959), pp. 253-255.
  20. J. F. DILLON, The generalized Langford-Skolem problem, Proc. 4th Southeastern Conference on Combinatorics, Graph Theory, Computing, Utilitas Math. Pub., Winnipeg, 1973, pp. 237- 247.
  21. P. ERDS AND R. L. GRAHAM, Old and new problems and results in combinatorial number theory, Monographies de l'Enseignement Math., to appear.
  22. P. ERD(SS AND H. HANANI, On a limit theorem in combinatorical analysis, Publ. Math. Debrecen, 10 (1963), pp. 10-13.
  23. P. ERD(S AND P. TURAN, On a problem of Sidon in additive number theory, and some related problems, J. London Math. Soc., 16 (1941), pp. 212-215 and 19 (1944), p. 208.
  24. R. FRUCHT, Graceful numbering of wheels and related graphs, Ann. N.Y. Acad. of Sci. 319 (1979), pp. 219-229.
  25. M. GARDNER, Mathematical games: the graceful graphs of Solomon Golomb, or how to number a graph parsimoniously, Scientific American, 226, 3 (1972), pp. 108-112; 226, 4 (1972), p. 104; 226, 6 (1972), p. 118.
  26. M. J. E. GOLAY, Note on the representation of 1, 2,..., n by differences, J. London Math. Soc., 4 (1972), pp. 729-734.
  27. S.W. GOLOMB, How to number a graph, in Graph Theory and Computing, R. C. Read, ed., Academic Press, New York, 1972, pp. 23-37.
  28. The largest graceful subgraph of the complete graph, Amer. Math. Monthly, 81 (1974), 499-501.
  29. S. W. GOLOMB AND G. S. BLOOM, Multi]arious applications of numbered graphs, in Proc. 2nd Caribbean Conference on Combinatorics and Computing, R. C. Read and C. C. Cadogan, eds., Univ. of West Indies, Cave Hill, Barbados, 1977, pp. 82-95.
  30. R. L. GRAHAM AND N. J. A. SLOANE, Lower bounds ]'or constant weight codes, IEEE Trans. Information Theory, IT-26 (1980), pp. 37-43.
  31. On constant weight codes and harmonious graphs, Utilitas Math., to appear.
  32. R. K. GuY, Monthly research problems 1969-73, Amer. Math. Monthly, 80 (1973), pp. 1120-1128.
  33. Monthly research problems 1969-75, ibid., 82 (1975), pp. 995-1004.
  34. Packing [1, n] with solutions of ax + by cz: the unity of combinatorics, in Teorie Combina- torie, Atti dei Convegni Lincei, 17, Vol. II, 1976, pp. 173-179.
  35. Monthly research problems 1969-77, Amer. Math. Monthly, 84 (1977), pp. 807-815, and 85 (1978), p. 263.
  36. R. K. GuY AND V. L. KEEL, JR., Monthly research problems 1969-71, Amer. Math. Monthly, 78 (1971), pp. 1113-1122.
  37. A. GYA.RFAS AND J. LEHEL, A method to generate graceful trees, Probl6mes Combinatoires et Th6orie des Graphs, Colloq. Intern. du Centre National de la Rech. Scient. 260, lditions du Centre National de la Recherche Scientifique, Paris, 1978, pp. 207-209.
  38. N. H.MMERER AND G. HOFMEISTER, Zu einer Vermutung yon Rohrbach, J. Reine Angew. Math. 286/287 (1976), pp. 239-247.
  39. E. HRTTER, Basen fiir Gitterpunktmengen, J. Reine Angew. Math. 202 (1959), pp. 153-170.
  40. G. HAJt3S, Sur la factorisation des groupes abgliens, (asopis Pst. Mat. Fys., 74 (1950), pp. 157-162.
  41. Sur le problkme de factorisation des groupes cycliques, Acta Math. Acad. Sci. Hungar., (1950), pp. 189-195.
  42. H. HALBERSTAM AND K. F. ROTH, Sequences, Vol. 1, Oxford University Press, Oxford, 1966.
  43. J. I. HALL, A. J. E. M. JANSEN, A. W. J. KOLEN AND J. H. VAN LINT, Equidistant codes with distance 12, Discrete Math., 17 (1977), pp. 71-83.
  44. H. HANANI, A note on Steiner triple systems, Math. Scand., 8 (1960), pp. 154-156.
  45. C. B. HASELGROVE AND J. LEECH, Note on restricted difference bases, J. London Math. Soc. 32 (1957), pp. 228-231.
  46. R. D. HEATH-BROWN AND H. IWANIEC, On the difference between consecutive primes, Bull Amer. Math. Soc., (1979), pp. 758-760.
  47. R. L. HEIMER AND H. LANGENBACH, The stamp problem, J. Recreational Math., 7 (1974), pp. 235-250.
  48. C. HOEDE AND H. KUIPER, All wheels are graceful, Utilitas Math., 14 (1978), p. 311.
  49. J. R. ISBELL, Perfect addition sets, Discrete Math., 24 (1978), pp. 13-18.
  50. W. KLOTZ, Line obere Schranke fiir die Reichweite einer Extremalbasis zweiter Ordnung, J. Reine Angew. Math., 238 (1969), pp. 161-168.
  51. K. M. KOH, T. TAN AND D. G. ROGERS, Two theorems on graceful trees, Discrete Math., 25 (1979), pp. 141-148.
  52. A. KOTZG, On certain vertex-valuations o[finite graphs, Utilitas Math., 4 (1973), pp. 261-290.
  53. A. KOTZIG AND A. ROSA, Magic valuations of finite graphs, Canad. Math. Bull., 13 (1970), pp. 451-461.
  54. A. KOTZIG AND J. TURGEON, -Valuations of regular graphs with complete components, in Combinatorics, A. Hajnal and V. T. S6s, eds., Colloq. Math. Soc. Jnos Bolyai, 18, North- Holland, Amsterdam, 2 vols., 1978, pp. 697-703.
  55. C. W. H. LAM, Nth power residue addition sets, J. Combinatorial Theory, 20A (1976), pp. 20-33.
  56. C. D. LANGFORD, Problem, Math. Gazette, 42 (1958), p. 228.
  57. J. LEECH, On the representation of 1, 2,. ., n by differences, J. London Math. Soc., 31 (1956), pp. 160-169.
  58. Another tree labelling problem, Amer. Math. Monthly, 82 (1975), pp. 923-925.
  59. E. LEVINE, On the generalized Langford problem, Fibonacci Quarterly, 6 (1968), pp. 135-138.
  60. W. F. LUNNON, A postage stamp problem, Computer J., 12 (1969), pp. 377-380.
  61. D. MCCARTHY, R. C. MULLIN, P. J. SCHELLENBERG, R. G. STANTON, AND S. A. VANSTONE, On approximations to a projective plane of order 6, Ars Combinatoria 2 (1976), pp. 111-168.
  62. F. J. MACWILLIAMS AND N. J. A. SLOANE, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
  63. M. MAHEO, Strongly graceful graphs, Discrete Math., 29 (1980), pp. 39-46.
  64. J. C. P. MILLER, Difference bases: three problems in additive number theory, in Computers in Number Theory, A. O. L. Atkin and B. J. Birch, eds., Academic Press, N.Y., 1971, pp. 299-322.
  65. L. MOSER, On the representation of 1, 2,..., n by sums, Acta Arith., 6 (1960), pp. 11-13.
  66. L. MOSER, J. R. POUNDER AND J. RIDDELL, On the cardinality o[ h-bases for n, J. London Math. Soc., 44 (1969), pp. 397-407.
  67. M. B. NATHANSON, Additive h-bases ]'or lattice points, Second International Conference on Combinatorial Mathematics, Ann. N.Y. Acad. Sci., 319 (1979), pp. 413-414.
  68. R. NOWAKOWSKI, Generalizations of the Lang]:ord-Skolem problem, M.Sc. Thesis, University of Calgary, 1975.
  69. C. J. PRIDAY, On Langford's problem (I), Math. Gazette, 43 (1959), pp. 250-253.
  70. S. P. RAO HEBBARE, Graceful cycles, Utilitas Math. 10 (1976), pp. 307-317.
  71. H. ROHRBACH, Ein Be#rag zur additiven Zahlentheorie, Math. Zeit., 42 (1937), pp. 1-30.
  72. Anwendung eines Satzes der additiven Zahlentheorie auf eine grappentheoretische Frage, ibid., pp. 538-542.
  73. A. ROSA, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N.Y. and Dunod, Paris, 1967, pp. 349-355.
  74. D. P. ROSELLE, Distributions of integers into s-tuples with given differences, Proc. Manitoba Conf. Number. Math., Dept. of Computer Science, University of Manitoba, Winnipeg, 1971, pp. 31-42.
  75. Comments and complements, Amer. Math. Monthly, 81 (1974), pp. 1097-1099.
  76. D. P. ROSELLE AND T. C. THOMASSON, JR, On generalized Langford sequences, J. Combinatorial Theory, 11A (1971), pp. 196-199.
  77. A.O. SANDS, On the factorisation offinite abelian groups, Acta Math. Acad. Sci. Hungar., 8 (1957), pp. 65-86 and 13 (1962), pp. 153-169.
  78. The factorization of Abelian groups, Quart. J. Math. Oxford (2), 10 (1959), pp. 81-91.
  79. J. SCH6NHEIM, On maximal systems of k-tuples, Stud. Sci. Math. Hungar., (1966), pp. 363-368.
  80. G. J. SIMMONS, Synch-sets: a variant of difference sets, Proc. 5th Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Math. Pub. Co., Winnipeg, 1974, pp. 625-645.
  81. T. SKOLEM, On certain distributions of integers'in pairs with given differences, Math. Scand., 5 (1957), pp. 57-68.
  82. R. G. STANTON, J. A. BATE AND R. C. MULLIN, Some tables for the postage stamp problem, Proc. Fourth Manitoba Conference on Numerical Math., Utilitas Math. Pub., Winnipeg, 1974, pp. 351-356.
  83. R. G. STANTON, J. G. KALBFLEISCH AND R. C. MULLIN, Covering and packing designs, Proc 2nd Chapel Hill Conference on Combinatorical Mathematics and its Applications, R. C. Bose et al., eds., Chapel Hill, 1970, pp. 428-450.
  84. R. G. STANTON AND C. R. ZARNKE, Labelling of balanced trees, Proc. 4th Southeastern Conference on Combinatorics, Graph Theory, Computing, Utilitas Math. Pub., Winnipeg, 1973, pp. 479-495.
  85. A. ST(3HR, Gel6ste und ungeliste Fragen fiber Basen der natiirlichen Zahlenreihe, J. Reine Angew. Math., 194 (1955), pp. 40-65 and 111-140.
  86. H. TAYLOR, Odd path sums in an edge-labeled tree, Math. Mag., 50 (1977), pp. 258-259.
  87. B. WICHMANN, A note on restricted difference bases, J. London Math. Soc., 38 (1962), pp. 465-466.

Related papers

A Study on Integer Additive Set-Graceful Graphs
K. A. Germina

2015

A set-labeling of a graph G is an injective function f:V(G)→P(X), where X is a finite set and a set-indexer of G is a set-labeling such that the induced function f^⊕:E(G)→P(X)-{∅} defined by f^⊕(uv) = f(u)⊕f(v) for every uv∈ E(G) is also injective. An integer additive set-labeling is an injective function f:V(G)→P(N_0), N_0 is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function f^+:E(G) →P(N_0) defined by f^+ (uv) = f(u)+ f(v) is also injective. In this paper, we extend the concepts of set-graceful labeling to integer additive set-labelings of graphs and provide some results on them.

View PDFchevron_right
The Exquisite Integer Additive Set-Labeling of Graphs
K. A. Germina

2015

Let N0 denote the set of all non-negative integers and P(N0) be its power set. An integer additive set-indexer (IASI) of a graph G is an injective function f:V(G) → P(N0) such that the induced function f :E(G) → P(N0) defined by f (uv) = f(u) + f(v) is also injective, where f(u) + f(v) is the sum set of f(u) and f(v). If f(uv) = k ∀ uv ∈ E(G), then f is said to be a k-uniform integer additive set-indexer. In this paper, we study the admissibility of a particular type of integer additive set-indexers by certain graphs.

View PDFchevron_right
Odd Harmonious Labeling of Some Classes of Graphs
P. Jeyanthi

Cubo (Temuco), 2020

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, • • • , 2q − 1} such that the induced function f * : E(G) → {1, 3, • • • , 2q − 1} defined by f * (uv) = f (u) + f (v) is a bijection. In this paper we prove that T p-tree, T•P m , T•2P m , regular bamboo tree, C n• P m , C n• 2P m and subdivided grid graphs are odd harmonious. RESUMEN Un grafo G(p, q) se dice impar armonioso si existe una inyección f : V (G) → {0, 1, 2, • • • , 2q − 1} tal que la función inducida f * : E(G) → {1, 3, • • • , 2q − 1} definida por f * (uv) = f (u) + f (v) es una biyección. En este artículo probamos que los grafos T p-árboles, T•P m , T•2P m ,árboles bambú regulares, C n• P m , C n• 2P m y cuadrículas subdivididas son impar armoniosos.

View PDFchevron_right
Results on Harmonious Graphs
Dushyant Tanna

This paper aims to focus on some of the results already in existence for the Harmonious Labeling of Graphs. The approach will be to summarize them in different flavor and possible different labeling. Several of these families of graphs were suggested by the application like consider a network of transmitting stations, each of which must be able to communicate with certain others-those to which it is linked in the network. The total bandwidth available is divided into e channels, where eis the number of links in the network, and each station x is assigned a number λ(x) fromℤ . When x and y communicate they use channel number λ(x) +λ (y). If the numbering is harmonious, each channel is assigned to exactly one link.Harmonious graphs are also interesting because they lead to modular versions of various combinatorial problems. For example, a harmonious labeling of the friendship graph Fnmay be regarded as a modular generalization of the Langford-Skolem Problem.

View PDFchevron_right
A study on the modular sumset labeling of graphs
Sudev Naduvath

Discrete Mathematics, Algorithms and Applications, 2017

For a positive integer [Formula: see text], let [Formula: see text] be the set of all non-negative integers modulo [Formula: see text] and [Formula: see text] be its power set. A modular sumset valuation or a modular sumset labeling of a given graph [Formula: see text] is an injective function [Formula: see text] such that the induced function [Formula: see text] defined by [Formula: see text]. A modular sumset indexer of a graph [Formula: see text] is an injective modular sumset valued function [Formula: see text] such that the induced function [Formula: see text] is also injective. In this paper, some properties and characteristics of this type of modular sumset labeling of graphs are being studied.

View PDFchevron_right

Related topics

  • Complete Graph
  • Complete Bipartite Graph

    • Cited by

    On Harmonious Labeling of Corona Graphs
    Martin Bača, Maged Youssef

    Journal of Applied Mathematics, 2014

    A graph with edges is said to be harmonious, if there is an injection from the vertices of to the group of integers modulo such that when each edge is assigned the label ( ) + ( ) (mod ), the resulting edge labels are distinct. In this paper, we study the existence of harmonious labeling for the corona graphs of a cycle and a graph and for the corona graph of 2 and a tree.

    View PDFchevron_right
    The Cordial Labeling for the Cartesian Product Between Paths and Cycles
    MOHAMMED AL-SHAMIRI

    International Journal of Research -GRANTHAALAYAH

    A graph is said to be cordial if it has a 0-1 labeling that satisfies certain properties. In this paper we show the Cartesian product of a path and a cycle or vice versa are always cordial under some conditions. Also, we prove that the Cartesian product of two paths is cordial.

    View PDFchevron_right
    Harmonious labeling for the corona graphs of small complete graph
    Denny R Silaban

    PROCEEDINGS OF THE 4TH INTERNATIONAL SYMPOSIUM ON CURRENT PROGRESS IN MATHEMATICS AND SCIENCES (ISCPMS2018)

    Supposed be a simple graph with a set of vertices (), set of edges (), with | ()| =. A graph is a harmonious graph if there is an injective function : () → ℤ , such that the induced function * : () → ℤ defined by * () = () + (), ∀ ∈ () is a bijective function. The function is called harmonious labeling of. It was known that a complete graph is harmonious only for ≤ 4 In this paper, we investigate the existence of harmonious labeling of the graphs from the corona operation between a complete graph 4 and , and also between 5 and

    View PDFchevron_right
    A Graceful Labeling of Square of Path Graph with Quadratic Complexity Algorithm
    eman abuhijleh

    WSEAS TRANSACTIONS ON MATHEMATICS, 2021

    A method for relaxed graceful labeling of P2n graphs is presented together with an algorithm designed for labeling these graphs. Graceful labeling is achieved by relaxing the range to 2m and perform the labeling using an algorithm with quadratic complexity (O(n2)). The algorithm can be used for labeling any P2n graph with n ≥ 3, as far as the machine can handle the size of the problem.

    View PDFchevron_right
    580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved