The wild number of an edge-colored graph
Combinatorics
2025-08-12 v1
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.
Cite
@article{arxiv.2508.06711,
title = {The wild number of an edge-colored graph},
author = {Katie Anders and Briana Foster-Greenwood and Rebecca Garcia and Naomi Krawzik},
journal= {arXiv preprint arXiv:2508.06711},
year = {2025}
}
Comments
22 pages, 9 figures, 2 tables