On the Neighbor Sum Distinguishing Index of Planar Graphs

Let c be a proper edge colouring of a graph G = (V,E) with integers 1, 2, . . . , k. Then k ≥ ∆(G), while by Vizing’s theorem, no more than k = ∆(G)+ 1 is necessary for constructing such c. On the course of investigating irregularities in graphs, it has beenmoreover conjectured that only slightly larger k, i.e., k = ∆(G) + 2 enables enforcing additional strong feature of c, namely that it attributes distinct sums of incident colours to adjacent vertices in G if only this graph has no isolated edges and is not isomorphic to C5. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph G of maximum degree at least 28which contains no isolated edges admits a proper edge colouring c : E → {1, 2, . . . ,∆(G) + 1} such that ∑