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arXiv · 2508.06711

The wild number of an edge-colored graph

Katie Anders·Briana Foster-Greenwood·Rebecca Garcia·Naomi Krawzik
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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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Katie Anders, Briana Foster-Greenwood, Rebecca Garcia, Naomi Krawzik. 2025-08-08. The wild number of an edge-colored graph. https://arxiv.org/abs/2508.06711

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