The wild number of an edge-colored graph

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.