Taylor's Condition

For a given Positive Integer , does there exist a Weighted Tree with Vertices whose paths have weights 1, 2, ..., , where is a Binomial Coefficient? Taylor showed that no such Tree can exist unless it is a Perfect Square or a Perfect Square plus 2. No such Trees are known except , 3, 4, and 6.

References

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1985.

Leech, J. ``Another Tree Labeling Problem.'' Amer. Math. Monthly 82, 923-925, 1975.

Taylor, H. ``Odd Path Sums in an Edge-Labeled Tree.'' Math. Mag. 50, 258-259, 1977.

• Perron's Theorem
• Persistence
• Persistent Number
• Persistent Process
• Perspective
• Perspective Axis
• Perspective Center
• Perspective Collineation
• Perspective Triangles
• Perspectivity
• Pesin Theory
• Petersen Graphs
• Petersen-Shoute Theorem
• Peters Projection
• Peter-Weyl Theorem
• Petrie Polygon
• Petrov Notation
• Pfaffian Form
• p-Good Path
• p-Group
• Phase
• Phase Space
• Phase Transition
• Phasor
• Phi Curve