### Distant sum distinguishing index of graphs with bounded minimum degree

#### Abstract

For any graph *G* = (*V*, *E*) with maximum degree Δ and without isolated edges, and a positive integer *r*, by χ′_{Σ, r}(*G*) we denote the *r*-distant sum distinguishing index of *G*. This is the least integer *k* for which a proper edge colouring *c*: *E* → {1, 2, …, *k*} exists such that ∑_{e∋u}*c*(*e*) ≠ ∑_{e∋v}*c*(*e*) for every pair of distinct vertices *u*, *v* at distance at most *r* in *G*. It was conjectured that χ′_{Σ, r}(*G*) ≤ (1 + *o*(1))Δ^{r − 1} for every *r* ≥ 3. Thus far it has been in particular proved that χ′_{Σ, r}(*G*) ≤ 6Δ^{r − 1} if *r* ≥ 4. Combining probabilistic and constructive approach, we show that this can be improved to χ′_{Σ, r}(*G*) ≤ (4 + *o*(1))Δ^{r − 1} if the minimum degree of *G* equals at least ln^{8} Δ.

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PDFDOI: https://doi.org/10.26493/1855-3974.1496.623

ISSN: 1855-3974

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