Regular graphs are universally 3-edge-weightable
Abstract
A graph is universally k-edge-weightable if for every k-element set Q⊂R, it admits a proper Q-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression \a,b,c\, every nice regular graph has a proper \a,b,c\-edge weighting. We prove that this remains valid for all 3-element set \a,b,c\ with c-b ≠ b-a. Consequently, every nice regular graph is universally 3-edge-weightable.
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