Neighbor Sum Distinguishing Coloring of Some Graphs

Abstract

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

On the Neighbor Sum Distinguishing Index of Planar Graphs

“…In particular it implies that for every planar graph G without an isolated edge. Independently, planar graphs were also investigated in and , where the bounds and , resp., were proved for these graphs. See also for other results concerning .…”

“…Independently, planar graphs were also investigated in and , where the bounds and , resp., were proved for these graphs. See also for other results concerning . The main result of this article, see Theorem , not only implies that Conjecture is valid for planar graphs of maximum degree at least 28, but also strengthens it by assuring sufficiency of an upper bound from Vizing's theorem to hold within our much more restrictive setting, that is, we prove that for these graphs.…”

Neighbor sum distinguishing edge colorings of sparse graphs

“…Then ndi Σ (G) ≤ ∆(G) + 3col(G) − 4. Theorem 1.7 (Dong and Wang [5]). Let G be a normal planar graph.…”

Neighbor Sum Distinguishing Edge Colorings of Graphs with Small Maximum Average Degree

“…Dong et al [12] considered the neighbor sum distinguishing colorings of planar graphs and showed that if G is a normal planar graph, then ndi (G) ≤ max{2 (G) + 1, 25}. In [13], Dong et al proved that if G is a normal graph and mad(G) ≤ 5/2, then ndi (G) ≤ k where k = max{ (G) + 1, 6}.…”