Graphs and Combinatorics

, Volume 29, Issue 4, pp 901–912 | Cite as

On Modular Edge-Graceful Graphs

  • Futaba Fujie-Okamoto
  • Ryan Jones
  • Kyle Kolasinski
  • Ping Zhang
Original Paper

Abstract

Let G be a connected graph of order \({n\ge 3}\) and size m and \({f:E(G)\to \mathbb{Z}_n}\) an edge labeling of G. Define a vertex labeling \({f': V(G)\to \mathbb{Z}_n}\) by \({f'(v)= \sum_{u\in N(v)}f(uv)}\) where the sum is computed in \({\mathbb{Z}_n}\) . If f′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where \({n\not\equiv 2 \pmod 4}\) , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where \({n\not\equiv 2\pmod 4}\) having diameter at most 5 is modular edge-graceful.

Keywords

Modular edge-graceful labeling Modular edge-graceful graph 

Mathematics Subject Classification (2000)

05C05 05C78 

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Copyright information

© Springer 2012

Authors and Affiliations

  • Futaba Fujie-Okamoto
    • 1
  • Ryan Jones
    • 2
  • Kyle Kolasinski
    • 2
  • Ping Zhang
    • 2
  1. 1.Mathematics DepartmentUniversity of Wisconsin La CrosseLa CrosseUSA
  2. 2.Department of MathematicsWestern Michigan UniversityKalamazooUSA

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