מושגי ליבה
The 25-vertex triangle-free graph is 3-dicritical, demonstrating the minimum size for a 3-dichromatic triangle-free graph.
תקציר
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.
Introduction
Definition of a 3-dicritical graph.
Overview of the study on the 25-vertex triangle-free graph.
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.
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.
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.
סטטיסטיקה
D25 = ⃗C←5 is a 3-dicritical oriented triangle-free graph on 25 vertices.
ציטוטים
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