Some bounds on the neighbor-distinguishing index of graphs

Abstract

a b s t r a c tA proper edge coloring of a graph G is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The neighbor-distinguishing index of G is the minimum number χ ′ a (G) of colors in a neighbor-distinguishing edge coloring of G.Let G be a graph with maximum degree ∆ and without isolated edges. In this paper, we prove that χ ′ a (G) ≤ 2∆ if 4 ≤ ∆ ≤ 5, and χ ′ a (G) ≤ 2.5∆ if ∆ ≥ 6. This improves a result in Zhang et al. (2014), which states that χ ′ a (G) ≤ 2.5∆ + 5 for any graph G without isolated edges. Moreover, we prove that if G is a semi-regular graph (i.e., each edge of G is incident to at least one ∆-vertex), then χ ′ a (G) ≤ 5 3 ∆ + 13 3 .

A Characterization for the Neighbor-Distinguishing Index of Planar Graphs

“…(1) − (3) has been proved in [13], and we only need to prove (4) and (5). If there are three consecutive vertices v i+1 , v i+2 , v i+3 ∈ N H 4,5 (v), then one of the faces incident with vv i+2 is at least a 4-face without 2-vertex or 5 + -face by Claim 10(3).…”

“…Conjecture 1 was confirmed by Balister et al [2] for bipartite graphs and graphs with ∆ = 3. Akbari et al [3] established that χ a (G) ≤ 3∆ for any normal graph G. This result is refined gradually to that of χ a (G) ≤ 2.5∆ + 5 in [4], to that of χ a (G) ≤ 2.5∆ in [5] and to that of χ a (G) ≤ 2∆ + 2 in [6]. By means of the probabilistic method, Hatami [7] confirmed that every normal graph G for a large enough ∆ has χ a (G) ≤ ∆ + 300.…”

Neighbor Distinguishing Colorings of Graphs with the Restriction for Maximum Average Degree

“…With the aid of the minimality of G, an NDEk-coloring ϕ of H exists through the use of the color set C in which case ϕ(vv p ) = p − 1 for p ∈ [2,5]. On the basis of Claim 1, we have s ≥ 2.…”

“…With the aid of the minimality of G, an NDT-k-coloring φ of H exists through the use of the color set C. In this case, φ(v) = 1 and φ(vv p ) = p for p ∈ [2,5]. The colors of the vertices v 1 , v 2 are removed.…”

“…Balister et al [2] verified that the conjecture above is true for graphs with ∆ = 3 and bipartite graphs. Akbari et al [3] provided an upper bound 3∆ for the neighbor distinguishing edge chromatic number χ ′ a (G) of an arbitrary graph G in terms of the maximum degree ∆ of G. By means of the edge-partitions of graphs, this upper bound was improved gradually to that of 2.5∆ + 5 due to Zhang et al [4], to that of 2.5∆ due to Wang et al [5] and to that of 2∆ + 2 due to Vučković [6]. Hatami [7] used the probabilistic method to establish that χ ′ a (G) ≤ ∆ + 300 for every graph G having a large enough ∆.…”