On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs
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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 \(n-2\).
KeywordsRainbow-cycle-forbidding Edge coloring Edge cut
Mathematics Subject Classification05C151
- 4.Fujita, S., Magnant, C., Ozeki, K.: Rainbow generalizations of Ramsey Theory—a dynamic survey. Theory Appl. Graphs (1), 1 (2014). https://doi.org/10.20429/tag2014.000101. http://digitalcommons.georgiasouthern.edu/tag/vol0/iss1/1(p. 38)