Neighbor Sum Distinguishing Index

Abstract

We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndi (G). In the paper we conjecture that for any connected graph G = C 5 of order n ≥ 3 we have ndi (G) ≤ (G) + 2. We prove this conjecture for several classes of graphs. We also show that ndi (G) ≤ 7 (G)/2 for any graph G with (G) ≥ 2 and ndi (G) ≤ 8 if G is cubic.

On the Total-Neighbor-Distinguishing Index by Sums

“…A similar conjecture by Przybyło and Woźniak states that two colors are enough by general total coloring [9]. Recently, some other authors in [3] considered a proper coloring of edges distinguishing adjacent vertices by sums.…”

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

“…Recently, Flandrin et al [11] studied the neighbor sum distinguishing colorings of cycles, trees, complete graphs, and complete bipartite graphs. Based on these examples, they proposed the following conjecture.…”

“…Based on these examples, they proposed the following conjecture. Conjecture 1.2 [11] If G is a connected graph on at least three vertices and G = C 5 , then ndi (G) ≤ (G) + 2.…”

Neighbor Sum Distinguishing Index of $$K_4$$ K 4 -Minor Free Graphs

“…Lemma 2 [11] If m = 0 (mod 3), then nsdi(C m ) = 3, otherwise, nsdi(C m ) = 4 with the exception that nsdi(C 5 ) = 5.…”

“…, k} and let φ : E(G) → C be a [k]-edge coloring of G. By m φ (v) and C φ (v) respectively, we denote the sum and the set of colors taken on the edges incident to v, i.e., m φ (v) = uv∈E(G) φ(uv) and C φ (v) = {φ(uv) | uv ∈ E(G)}. If the coloring φ is proper and satisfies that m φ (v) = m φ (u) for each edge uv ∈ E(G), then we call such coloring a neighbor sum distinguishing [k]-edge coloring of G. By nsdi(G) , we denote the smallest value k such that G has a neighbor sum distinguishing [k]-edge coloring of G. This concept was first introduced by Flandrin et al [11]. Similarly, Zhang et al [23] has introduced the following concept.…”