The 1,2-Conjecture for powers of cycles

A [k]-total-weighting ω of a simple graph G is a mapping ω:V(G)∪E(G)→{1,…,k}. A [k]-total-weighting ω of G is neighbour-distinguishing if, for each pair of adjacent vertices u,v∈V(G), the value ω(u)+∑uw∈E(G)ω(uw) is distinct from ω(v)+∑vw∈E(G)ω(vw). The 1,2-Conjecture states that every simple graph G has a neighbour-distinguishing [2]-total-weighting. In this work, we prove that the 1,2-Conjecture is valid for all powers of cycles.