Distant sum distinguishing index of graphs with bounded minimum degree

Abstract

For any graph G = (V, E) with maximum degree ∆ 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))∆ r−1 for every r ≥ 3. Thus far it has been in particular proved that χ ′ Σ,r (G) ≤ 6∆ 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))∆ r−1 if the minimum degree of G equals at least ln 8 ∆. Keywords: distant sum distinguishing index of a graph, neighbour sum distinguishing index, adjacent strong chromatic index, distant set distinguishing index d c (v) := e∋v c(e).