On the neighbour sum distinguishing index of planar graphs

Abstract

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\dots ,k$. Then $k\ge \Delta (G)$, while by Vizing's theorem, no more than $k=\Delta (G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=\Delta (G)+2$ 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 $C_5$. 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 $28$ which contains no isolated edges admits a proper edge colouring $c:E\to \{1,2,\dots ,\Delta (G)+1\}$ such that ${\sum }_{e\ni u}c(e)\ne {\sum }_{e\ni v}c(e)$ for every edge $uv$ of $G$.

22 pages

Cited by in corpus (5)

• Progress on the adjacent vertex distinguishing edge colouring conjecture
• A note on asymptotically optimal neighbour sum distinguishing colourings
• On the neighbour sum distinguishing index of graphs with bounded maximum average degree
• Distant sum distinguishing index of graphs
• Distant set distinguishing edge colourings of graphs