On Rainbow-cycle-forbidding Edge Colorings of Finite Graphs
Publication Date
10-1-2019
Document Type
Article
Organizational Units
Mathematics
Keywords
Rainbow-cycle-forbidding, Edge-colored, Finite graphs
Abstract
It is shown that whenever the edges of a connected simple graph on n vertices are colored with n−1" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">n−1n−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" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">n−1n−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" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">n−1n−1 replaced by n−2" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">n−2n−2.
Publication Statement
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Recommended Citation
Hoffman, Dean, et al. “On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs.” Graphs and Combinatorics, vol. 35, no. 6, 2019, pp. 1585–1596. doi: 10.1007/s00373-019-02102-6.