Toward a graph version of Rado's theorem
Andy Parrish
TL;DR
This paper introduces necessary and sufficient conditions for an equation to be graph-regular, meaning it always has monochromatic solutions in edge-colored complete graphs on natural numbers, extending Rado's theorem to a graph setting.
Contribution
It establishes two Rado-like conditions that characterize graph-regular equations, advancing the understanding of monochromatic solutions in graph colorings.
Findings
Identifies necessary and sufficient conditions for graph-regularity.
Extends Rado's theorem to a graph-theoretic context.
Provides a framework for analyzing monochromatic solutions in edge-colored graphs.
Abstract
An equation is called graph-regular if it always has monochromatic solutions under edge-colorings of the complete graph on the naturals. We present two Rado-like conditions which are respectively necessary and sufficient for an equation to be graph-regular.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph theory and applications