The 1-2-3 Conjecture almost holds for regular graphs

Jakub Przybyło

Published 2018-09-27Version 1

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set $\{1,2,3,4,5\}$. We prove that for regular graphs it is sufficient to use weights $1$, $2$, $3$, $4$.