Adjacent Vertex Distinguishing Edge‐Colorings

Abstract

An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χ a (G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χ a (G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χ a (G) ≤ Δ(G) + 2 for bipartite graphs. These bounds are tight. For k-chromatic graphs G without isolated edges we prove a weaker result of the form χ a (G) = Δ(G) + O(log k).

Neighbor Sum Distinguishing Index

“…It is easy to see that ndi(C 5 ) = 5 and in [8] it is conjectured that ndi(G) ≤ (G)+2 for any connected graph G = C 5 on n ≥ 3 vertices. The conjecture has been confirmed by Balister et al [3] for bipartite graphs and for graphs G with (G) = 3. Edwards et al [5] have shown even that ndi(G) ≤ (G) + 1 if G is bipartite, planar, and of maximum degree (G) ≥ 12.…”

“…On the other hand, by [3], ndi(G) ≤ 5 for these graphs. Using the concept of maximal (for a given n) Sidon sets, S n (that is any maximal size subset of [n] whose each pair of distinct elements has a unique sum, i.e., a…”

“…Proof Given any proper 5-edge coloring distinguishing the neighbors by sets, whose existence for cubic graphs was proven in [3], it is enough to use the colors from the set S 8 (S 8 = {1, 2, 3, 5, 8}) instead of the original ones. The obtained proper [8]-coloring v of G is nsd, since S(x) ∩ S(y) = {v(x y)} and S(x) \ {v(x y)} = S(y) \ {v(x y)} whenever vertices x, y are adjacent in G.…”

On Twin Edge Colorings of d Infinite Paths

“…Recently, an adjacent vertex distinguishing edge-coloring [1] of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices has the same set of colors. The minimum number of colors of G is the adjacent vertex-distinguishing chromatic number, denoted by…”

“…In 2006, Baril [2] show that the adjacent vertex-distinguishing chromatic number of the multidimensional mesh and the hypercube both are equal to the maximum degree of the both graphs plus one. In 2007, Balister [1] prove for bipartite graphs or such graphs with maximum degree…”