A Characterization on the Adjacent Vertex Distinguishing Index of Planar Graphs with Large Maximum Degree

2015

Discrete Mathematics

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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 .

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Section: Conjecturementioning

confidence: 98%

Some bounds on the neighbor-distinguishing index of graphs

2015

Discrete Mathematics

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“…This was proved in [9] for ∆ ≥ 12 and in [10] for ∆ = 11. Furthermore, in 2015, Wang and Huang [11] gave the following stronger results:…”

Section: Introductionmentioning

confidence: 99%

“…Assume to the contrary that k ≤ 10. Firstly, vv 1 or vv 2 is recolored with any color in [11,13]. Secondly, we recolor vv 1 with 11 and vv 2 with 12 or 13.…”

mentioning

confidence: 99%

“…a ∈ [1, 10]. Firstly, we recolor vv 1 with any color in [11,13]. Secondly, we exchange the colors of vv 2 and v 2 u 1 , and recolor vv 1 with any color in [11,13].…”

mentioning

confidence: 99%

“…Firstly, we recolor vv 1 with any color in [11,13]. Secondly, we exchange the colors of vv 2 and v 2 u 1 , and recolor vv 1 with any color in [11,13]. Thirdly, for each v i with d(v i ) ≤ 3 and i ≥ 3, we recolor vv i with a color b i ∈ {1, 11 − 13} such that v i does not conflict with the vertices adjacent to v i , and vv 1 with any color in [11,13] \ {b i }.…”

mentioning

confidence: 99%

See 2 more Smart Citations

A Characterization for the Neighbor-Distinguishing Index of Planar Graphs

2022

Symmetry

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index χa′(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G≠C5, then χa′(G)≤Δ+2. In this paper, we show that if G is a planar graph with maximum degree Δ≥13, then Δ≤χa′(G)≤Δ+1, and, further, χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree.

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Neighbor Sum Distinguishing Index of Subcubic Graphs

Graphs and Combinatorics

2017