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2012 | 32 | 3 | 487-505
Tytuł artykułu

Nowhere-zero modular edge-graceful graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For a connected graph G of order n ≥ 3, let f: E(G) → ℤₙ be an edge labeling of G. The vertex labeling f': V(G) → ℤₙ induced by f is defined as $f'(u) = ∑_{v ∈ N(u)} f(uv)$, where the sum is computed in ℤₙ. If f' is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) ≠ 0 for all e ∈ E(G) and in this case, G is a nowhere-zero modular edge-graceful graph. It is shown that a connected graph G of order n ≥ 3 is nowhere-zero modular edge-graceful if and only if n ≢ 2 mod 4, G ≠ K₃ and G is not a star of even order. For a connected graph G of order n ≥ 3, the smallest integer k ≥ n for which there exists an edge labeling f: E(G) → ℤₖ - {0} such that the induced vertex labeling f' is one-to-one is referred to as the nowhere-zero modular edge-gracefulness of G and this number is determined for every connected graph of order at least 3.
Wydawca
Rocznik
Tom
32
Numer
3
Strony
487-505
Opis fizyczny
Daty
wydano
2012
otrzymano
2011-06-01
poprawiono
2011-09-04
zaakceptowano
2011-09-04
Twórcy
autor
  • Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
autor
  • Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
Bibliografia
  • [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal and B.A. Reed, Vertex colouring edge partitions, J. Combin. Theory (B) 94 (2005) 237-244, doi: 10.1016/j.jctb.2005.01.001.
  • [2] P.N. Balister, E. Györi, J. Lehel and R.H. Schelp, Adjacent vertex distinguishing edge-colorings, SIAM J. Discrete Math. 21 (2007) 237-250, doi: 10.1137/S0895480102414107.
  • [3] G. Chartrand, L. Lesniak and P. Zhang, Graphs & Digraphs ( Fifth Edition, Chapman & Hall/CRC, Boca Raton, FL, 2010).
  • [4] G. Chartrand, F. Okamoto and P. Zhang, The sigma chromatic number of a graph, Graphs Combin. 26 (2010) 755-773, doi: 10.1007/s00373-010-0952-7.
  • [5] G. Chartrand and P. Zhang, Chromatic Graph Theory (Chapman & Hall/CRC Press, Boca Raton, 2009).
  • [6] E. Györi, M. Horňák, C. Palmer and M. Woźniak, General neighbor-distinguishing index of a graph, Discrete Math. 308 (2008) 827-831, doi: 10.1016/j.disc.2007.07.046.
  • [7] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 16 (2009) #DS6.
  • [8] R.B. Gnana Jothi, Topics in Graph Theory, Ph.D. Thesis, Madurai (Kamaraj University, 1991).
  • [9] S.W. Golomb, How to number a graph, Graph Theory and Computing, (Academic Press, New York, 1972) 23-37.
  • [10] R. Jones, K. Kolasinski, F. Okamoto and P. Zhang, Modular neighbor-distinguishing edge colorings of graphs, J. Combin. Math. Combin. Comput. 76 (2011) 159-175.
  • [11] R. Jones, K. Kolasinski and P. Zhang, A proof of the modular edge-graceful trees conjecture, J. Combin. Math. Combin. Comput., to appear.
  • [12] S.P. Lo, On edge-graceful labelings of graphs, Congr. Numer. 50 (1985) 231-241.
  • [13] F. Okamoto, E. Salehi and P. Zhang, A checkerboard problem and modular colorings of graphs, Bull. Inst. Combin Appl. 58 (2010) 29-47.
  • [14] F. Okamoto, E. Salehi and P. Zhang, A solution to the checkerboard problem, Intern. J. Comput. Appl. Math. 5 (2010) 447-458.
  • [15] A. Rosa, On certain valuations of the vertices of a graph in: Theory of Graphs, Proc. Internat. Sympos. Rome, 1966 (Gordon and Breach, New York, 1967) 349-355.
  • [16] Z. Zhang, L. Liu and J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626, doi: 10.1016/S0893-9659(02)80015-5.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1621
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