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19 pages, 347 KiB

Open AccessArticle

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

by Jingjing Huo, Sensen Wen, Yulong Chen and Mingchao Li

Axioms 2023, 12(12), 1132; https://doi.org/10.3390/axioms12121132 - 18 Dec 2023

Cited by 2 | Viewed by 1624

Neighbor distinguishing colorings of graphs represent powerful tools for solving the channel assignment problem in wireless communication networks. They consist of two forms of coloring: neighbor distinguishing edge coloring, and neighbor distinguishing total coloring. The neighbor distinguishing edge (total) coloring of a graph G is an edge (total) coloring with the requirement that each pair of adjacent vertices contains different color sets. The neighbor distinguishing edge (total) chromatic number of G is the smallest integer k in cases where a neighbor distinguishing edge (total) coloring exists through the use of k colors in G. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs. In this paper, we characterize the neighbor distinguishing edge (total) chromatic numbers of graphs with a maximum average degree less than four by means of the discharging method. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

17 pages, 857 KiB

Open AccessArticle

A Characterization for the Neighbor-Distinguishing Index of Planar Graphs

by Jingjing Huo, Mingchao Li and Ying Wang

Symmetry 2022, 14(7), 1289; https://doi.org/10.3390/sym14071289 - 21 Jun 2022

Cited by 3 | Viewed by 1586

Abstract

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

χ_{a}^{'} (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 \neq C_{5}

, then

χ_{a}^{'} (G) \leq Δ + 2

. In this paper, we show that if G is a planar graph with maximum degree

Δ \geq 13

, then

Δ \leq χ_{a}^{'} (G) \leq Δ + 1

, and, further,

χ_{a}^{'} (G) = Δ + 1

if and only if G contains two adjacent vertices of maximum degree. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

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11 pages, 285 KiB

Open AccessArticle

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ_2,1,2

by Donghan Zhang

Mathematics 2021, 9(7), 708; https://doi.org/10.3390/math9070708 - 25 Mar 2021

Cited by 11 | Viewed by 1731

Abstract

A theta graph

Θ_{2, 1, 2}

is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring

ϕ

of G is a proper total coloring [...] Read more.

A theta graph

Θ_{2, 1, 2}

is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring

ϕ

of G is a proper total coloring of G such that

\sum_{z \in E_{G} (u) \cup {u}} ϕ (z) \neq \sum_{z \in E_{G} (v) \cup {v}} ϕ (z)

for each edge

u v \in E (G)

, where

E_{G} (u)

denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree

Δ

admits an NSD total

(Δ + 3)

-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree

Δ \geq 9

but without theta graphs

Θ_{2, 1, 2}

by applying the Combinatorial Nullstellensatz, which improves the result of Song et al. Full article

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23 pages, 9185 KiB

Open AccessArticle

Gravitation-Based Edge Detection in Hyperspectral Images

by Genyun Sun, Aizhu Zhang, Jinchang Ren, Jingsheng Ma, Peng Wang, Yuanzhi Zhang and Xiuping Jia

Remote Sens. 2017, 9(6), 592; https://doi.org/10.3390/rs9060592 - 11 Jun 2017

Cited by 27 | Viewed by 8243

Abstract

Edge detection is one of the key issues in the field of computer vision and remote sensing image analysis. Although many different edge-detection methods have been proposed for gray-scale, color, and multispectral images, they still face difficulties when extracting edge features from hyperspectral images (HSIs) that contain a large number of bands with very narrow gap in the spectral domain. Inspired by the clustering characteristic of the gravitational theory, a novel edge-detection algorithm for HSIs is presented in this paper. In the proposed method, we first construct a joint feature space by combining the spatial and spectral features. Each pixel of HSI is assumed to be a celestial object in the joint feature space, which exerts gravitational force to each of its neighboring pixel. Accordingly, each object travels in the joint feature space until it reaches a stable equilibrium. At the equilibrium, the image is smoothed and the edges are enhanced, where the edge pixels can be easily distinguished by calculating the gravitational potential energy. The proposed edge-detection method is tested on several benchmark HSIs and the obtained results were compared with those of four state-of-the-art approaches. The experimental results confirm the efficacy of the proposed method. Full article

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