r-Strong edge colorings of graphs

Abstract

Let G be a graph and for any natural number r, s (G, r) denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, s (T , 1) (T ) + 1. Here we generalize this result and show that s (T , 2) (T ) + 1. Moreover, we show that if for any two vertices u and v with maximum degree d(u, v) 3, then s (T , 2) = (T ). Also for any tree T with (T ) 3 we prove that s (T , 3) 2 (T ) − 1. Finally, it is shown that for any graph G with no isolated edges, s (G, 1) 3 (G).

Neighbor Sum Distinguishing Index

“…The last theorem is weaker than the results of of Akbari et al [1] and Hatami [6] for graphs with large maximum degree.…”

Adjacent vertex distinguishing colorings by sum of sparse graphs

“…Hatami [11] showed that if G is a normal graph and ∆(G) > 10 20 , then χ ′ a (G) ≤ ∆(G) + 300. For more references, see [1,3,8,12]. Recently, in [13], Hocquard et al proved the following result.…”

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

“…1. By our assumption, we know that H 1 has an nsd- 3 ). Then we properly color vv 2 and vv 3 such that τ (vv 2 …”