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Graphs and Combinatorics

, Volume 35, Issue 6, pp 1585–1596 | Cite as

On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs

  • Dean Hoffman
  • Paul Horn
  • Peter Johnson
  • Andrew OwensEmail author
Original Paper
  • 57 Downloads

Abstract

It is shown that whenever the edges of a connected simple graph on n vertices are colored with \(n-1\) colors appearing so that no cycle in G is rainbow, there must be a monochromatic edge cut in G. From this it follows that such colorings of G can be represented, or ‘encoded,’ by full binary trees with n leaves, with vertices labeled by subsets of V(G), such that the leaf labels are singletons, the label of each non-leaf is the union of the labels of its children, and each label set induces a connected subgraph of G. It is also shown that \(n-1\) is the largest integer for which the main theorem holds, for each n, although for some graphs a certain strengthening of the hypothesis makes the theorem conclusion true with \(n-1\) replaced by \(n-2\).

Keywords

Rainbow-cycle-forbidding Edge coloring Edge cut 

Mathematics Subject Classification

05C151 

Notes

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA
  2. 2.Depatrtment of MathematicsUniversity of DenverDenverUSA

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