#### Title

On Rainbow-Cycle-Forbidding Edge Colorings of Finite Graphs

#### Document Type

Article

#### Publication Date

10-1-2019

#### Keywords

Rainbow-cycle-forbidding, Edge-colored, Finite graphs

#### Organizational Units

College of Natual Science and Mathematics, Mathematics

#### 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

Copyright held by author or publisher. User is responsible for all copyright compliance.

#### 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.