On the Neighbor Sum Distinguishing Index of Planar Graphs

Abstract

Let c be a proper edge coloring of a graph G=(V,E) with integers 1,2,…,k. Then k≥Δ(G), while Vizing's theorem guarantees that we can take k≤Δ(G)+1. On the course of investigating irregularities in graphs, it has been conjectured that with only slightly larger k, that is, k=Δ(G)+2, we could enforce an additional strong feature of c, namely that it attributes distinct sums of incident colors to adjacent vertices in G if only this graph has no isolated edges and is not isomorphic to C5. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact an even stronger statement holds, as the necessary number of colors stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph G of maximum degree at least 28, which contains no isolated edges admits a proper edge coloring c:E→{1,2,…,Δ(G)+1} such that ∑e∋ucfalse(efalse)≠∑e∋vcfalse(efalse) for every edge uv of G.

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

“…Now we exchange the colors of y 2 y 3 and y 3 y 4 , and color y 1 y 2 with a color in [5]\(φ(y 1 y 4 ) ∪ φ(y 2 y 3 ) ∪ φ(y 3 y 4 ) ∪ {φ(y 2 y 3 ) + φ(y 3 w)}), then we have an nsd- [5]-coloring of G, which is a contradiction.…”

“…If φ(y 1 y 4 ) = φ(y 2 y 3 ), since φ(y 2 y 3 ) ∈ C φ (y 1 ), we know that φ(y 1 u 1 ) = φ(y 2 y 3 ). We also have that φ(y 3 y 4 ) = α or else we color y 1 y 2 with φ(y 3 y 4 ), we will get an nsd- [5]-coloring of G. Since |F φ (y 1 y 2 )| = 5, it means that…”

“…Let z 1 denote the neighbor of z different from x, y. Let It is easy to see that max{ [5]\F ϕ (x y)} < m ϕ (z). Hence any coloring of x y using a color from [5]\F ϕ (x y) and any coloring of xz using …”

“…Then m φ (y 2 ) = m φ (y 1 ). We may assume that α = φ(y 3 y 4 ), otherwise we color y 1 y 2 with φ(y 3 y 4 ), then we will get an nsd- [5]coloring of G. Notice that F φ (y 1 y 2 ) = {φ(y 1 u 1 ), φ(y 2 y 3 ), φ(y 3 y 4 ), φ(y 3 y 4 ) − φ(y 1 u 1 ), φ(y 3 y 4 ) + φ(y 3 w)} = [5].…”

“…Wang et al [18] proved that nsdi(G) ≤ max{Δ + 10, 25} holds for normal planar graphs. Recently, Bonamy and Przybyło [5] proved the following theorem.…”