The wild number of an edge-colored graph
Katie Anders, Briana Foster-Greenwood, Rebecca Garcia, and Naomi Krawzik

TL;DR
This paper introduces the wild number, a new measure of how close an edge-colored graph is to having a spanning tree in every color, explores its computational complexity, and provides exact values for specific graph classes.
Contribution
It defines the wild number, proves NP-completeness of computing it, and determines exact wild numbers for trees, cycles, and certain restricted graphs.
Findings
Determining the wild number is NP-complete.
Bounds on the wild number are established.
Exact wild numbers are found for specific graph classes.
Abstract
We introduce the wild number of an edge-colored graph as a measure of how close an edge-colored graph is to having a spanning tree in every color. This combinatorial concept originates in the algebraic theory of generalized graph splines. After showing that determining the wild number of a graph is an NP-complete problem, we provide bounds on the wild number and find the exact wild number for trees, cycles, and families of graphs with restrictions on the edge-colorings. This article serves as an invitation to the topic of wild numbers and includes several open problems, many of which are suitable for undergraduate research projects.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Limits and Structures in Graph Theory · Advanced Graph Theory Research
