Discussiones Mathematicae Graph Theory 32(3) (2012) 487-505
doi: 10.7151/dmgt.1621

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Nowhere-zero Modular Edge-graceful Graphs

Ryan Jones and Ping Zhang

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA


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.

Keywords: modular edge-graceful labelings and graphs, nowhere-zero labelings, modular edge-gracefulness

2010 Mathematics Subject Classification: 05C05, 05C15, 05C78.


[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.

Received 1 June 2011
Revised 4 September 2011
Accepted 4 September 2011