The irregularity strength of \(tP_ 3\) (Q1182742)

The irregularity strength \(s(G)\) of a simple graph \(G\) is the smallest number such that the edges of \(G\) may be assigned weights \(\leq s(G)\) in such a way as to obtain distinct weight sums at each vertex. It is shown by elementary arguments in additive number theory that for \(G=tP_ 3\), the disjoint union of \(t\) paths of length 3, \([(15t-1)/7]+1\leq s(G)\leq [(15t-1)/7]+2\) (\([x]\) indicating the integer part of \(x\)). For \(t\equiv 3(\mod 7)\) the lower bound is reached, while this is conjectured to always be the case.