A note on asymptotically optimal neighbour sum distinguishing colourings

Abstract

The least k admitting a proper edge colouring c : E → {1, 2, . . . , k} of a graph G = (V, E) without isolated edges such that e∋u c(e) = e∋v c(e) for every uv ∈ E is denoted by χ ′ Σ (G). It has been conjectured that χ ′ Σ (G) ≤ ∆ + 2 for every connected graph of order at least three different from the cycle C 5 , where ∆ is the maximum degree of G. It is known that χ ′ Σ (G) = ∆ + O(∆ 5 6 ln 1 6 ∆) for a graph G without isolated edges. We improve this upper bound to χ ′ Σ (G) = ∆ + O(∆ 1 2 ) using a simpler approach involving a combinatorial algorithm enhanced by the probabilistic method. The same upper bound is provided for the total version of this problem as well.

Distant sum distinguishing index of graphs

“…where d(u, v) denotes the distance of u and v in G. This is called the r-distant sum distinguishing index of G. Such concept develops the study on the earlier neighbour sum distinguishing index of G, χ ′ Σ (G) = χ ′ Σ,1 (G), for which it was conjectured in [16] that χ ′ Σ (G) ≤ ∆(G) + 2 for any connected graph G of order at least three different from the cycle C 5 . This was asymptotically confirmed in [32] and [31], where it was showed that…”

Distant sum distinguishing index of graphs with bounded minimum degree

“…Upper bounds of orders conjectured above are also known for r = 2, 3, but with slightly worse multiplicative constants than in Theorem 1.2 above, see [31], while the upper bound of the form χ Σ (G) ≤ (1 + o(1))∆(G) was proved in [29] and [32], see also [6,11,14,28,33,34] for other results concerning the case r = 1. In this paper we combine probabilistic approach with a special constructive algorithm in order to provide the following improvements of the best known upper bounds for all r ≥ 4 from Theorem 1.2, under assumption that the minimum degree of a graph is larger than some poly-logarithmic function of the maximum degree.…”

“…< 10 48 • 10 64 ∆32 • 33∆ −16(r−1) < 10 114 ∆ −4r−12(r−4) Now for each vertex v ∈ C of degree d in G and every integer t ∈ [0, Q − 1] which is not congruent to 0 modulo 3, let X v,t denote (the random variable expressing) the number of vertices u in N r C (v) with d c (u) ∈ [t−31•93, t+31…”

Distant sum distinguishing index of graphs with bounded minimum degree

“…Then χ ′ Σ,r (G) ≤ 6∆ r−1 . Upper bounds of orders conjectured above are also known for r = 2, 3, but with slightly worse multiplicative constants than in Theorem 2 above, see [29], while the upper bound of the form χ ′ Σ (G) ≤ (1 + o(1))∆(G) was proved in [26] and [25], see also [6,11,14,32,33,34] for other results concerning the case r = 1. In this paper we combine probabilistic approach with a special constructive algorithm in order to provide the following improvements of the best known upper bounds for all r ≥ 4 from Theorem 2, under assumption that the minimum degree of a graph is larger than some poly-logarithmic function of the maximum degree.…”