Distant set distinguishing edge colourings of graphs (1508.05024v1)

Abstract: We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distant at most $r$ in a graph are called $r$-adjacent. The least number of colours in a proper edge colouring of a graph $G$ such that the sets of colours met by any $r$-adjacent vertices in $G$ are distinct is called the $r$-adjacent strong chromatic index of $G$ and denoted by $\chi'{a,r}(G)$. It has been conjectured that $\chi'{a,1}(G)\leq\Delta+2$ if $G$ is connected of maximum degree $\Delta$ and non-isomorphic to $C_5$, while Hatami proved that there is a constant $C$, $C\leq 300$, such that $\chi'{a,1}(G)\leq\Delta+C$ if $\Delta>10^{20}$ [J. Combin. Theory Ser. B 95 (2005) 246--256]. We conjecture that a similar statement should hold for any $r$, i.e., that for each positive integer $r$ there exist constants $\delta_0$ and $C$ such that $\chi'{a,r}(G) \leq \Delta+C$ for every graph without an isolated edge and with minimum degree $\delta \geq \delta_0$, and argue that a lower bound on $\delta$ is unavoidable in such a case (for $r>2$). Using the probabilistic method we prove such upper bound to hold for graphs with $\delta\geq \epsilon\Delta$, for every $r$ and any fixed $\varepsilon\in(0,1]$, i.e., in particular for regular graphs. We also support the conjecture by proving an upper bound $\chi'_{a,r}(G) \leq (1+o(1))\Delta$ for graphs with $\delta\geq r+2$.