Core Concepts
The 25-vertex triangle-free graph is 3-dicritical, demonstrating the minimum size for a 3-dichromatic triangle-free graph.
Abstract
The content discusses the construction and analysis of a 25-vertex triangle-free graph to demonstrate its 3-dicritical nature. It explores the implications of the graph's structure on its coloring properties, showcasing its uniqueness and minimum size for a 3-dichromatic triangle-free graph.
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Introduction
- Definition of a 3-dicritical graph.
- Overview of the study on the 25-vertex triangle-free graph.
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Construction of the Graph
- Description of the directed linear forest structure of the graph.
- Explanation of the packs and matched cycles within the graph.
- Illustration of the types of directed cycles present in the graph.
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Proof of 3-Dicritical Nature
- Analysis of potential monochromatic packs and cycles in the graph.
- Examination of the color distribution within the packs to identify contradictions.
- Demonstration of the impossibility of a 2-dicolouring for the 25-vertex triangle-free graph.
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Conclusion
- Confirmation of the 25-vertex triangle-free graph's 3-dicritical status.
- Discussion on the uniqueness and significance of the graph in graph coloring studies.
Stats
D25 = ⃗C←5 is a 3-dicritical oriented triangle-free graph on 25 vertices.
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