August 17, 2025Open Access

Rainbow numbers in planar host graphs

Key Points

The rainbow number is critical for understanding colorings of graphs, specifically in planar contexts.
Exact evaluations for small subgraphs like wheels and sunflowers have been determined, enhancing graph theory knowledge.
The analysis involves planar host graphs, particularly focusing on configurations like triangulations and wheels.
Understanding rainbow numbers in graphs could lead to applications in areas like network design and combinatorics.

Abstract

If F is a nonempty set of graphs that contain H as a subgraph, then the rainbow number rb (F, H) is the least t? N such that every t-coloring of F? F that uses all t colors contains a rainbow subgraph isomorphic to H. In this paper, we consider rainbow numbers when F = F where F is a wheel, a sunflower, or a double-hubbed wheel. Several exact evaluations are determined for various small subgraphs. Implications involving the case where F consists of all plane triangulations of order n are also discussed.

Rainbow numbers in planar host graphs

Key Points

Abstract

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