An upper bound for the harmonious chromatic number of a graph

Lee, Sin-Min; Mitchem, John

doi:10.1002/jgt.3190110414

Cited by 28 publications

(6 citation statements)

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“…Corollary 6. Any graph of order n having maximum degree n − 1 has harmonious chromatic number n. Lee and Mitchem (2006) present an upper bound for the harmonious chromatic number of a graph.…”

Section: Known Upper and Lower Bounds For The Harmonious Chromatic Nu...mentioning

confidence: 99%

Approximate and exact results for the harmonious chromatic number

Marinescu-Ghemeci¹,

Obreja²,

Popa³

2021

Preprint

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Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system.The results presented in this paper fall into two categories: in the first part of the paper we are concerned with the computational aspects of finding a minimum harmonious coloring and in the second part we determine the exact value of the harmonious chromatic number for some particular graphs and classes of graphs. More precisely, in the first part we show that finding a minimum harmonious coloring for arbitrary graphs is APX-hard, the natural greedy algorithm is a Ω( √ n)-approximation, and, moreover, we show a relationship between the vertex cover and the harmonious chromatic number. In the second part we determine the exact value of the harmonious chromatic number for all 3-regular planar graphs of diameter 3, some non-planar regular graphs and cycle-related graphs.

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Section: Known Upper and Lower Bounds For The Harmonious Chromatic Nu...mentioning

confidence: 99%

Approximate and exact results for the harmonious chromatic number

Marinescu-Ghemeci¹,

Obreja²,

Popa³

2021

Preprint

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“…The Harmonious chromatic number was introduced by Frank, Harary and Plantholt [40] and independently by Hoperoft and Krishnamoorthy [59]. These parameters were studied in detail by Lee and Mitchem [67] in 1987, by Mitchem [75] in 1989, by Miller and Pritikin [74] in 199l and by Kundrik [67] in1992.…”

Section: B a Survey Of Various Coloring Parametersmentioning

confidence: 99%

Graph coloring Parameters-A Survey

Kumar¹

2019

IJRASET

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“…O problema de Colorac ¸ão de Vértices consiste em determinar χ(G), para um dado grafo G. Se c é uma colorac ¸ão de um grafo G, para todos u, v ∈ V (G), definimos a cor de uma aresta e = uv ∈ E(G) como c(e) = {c(u), c(v)}. Dizemos que uma k-colorac ¸ão é harmônica ou linha-distinguível (definida com a terminologia line-distinguishing em [Lee and Mitchem 1987]) quando, para quaisquer duas arestas distintas e 1 e e 2 , temos c(e 1 ) ̸ = c(e 2 ). Note que tal colorac ¸ão não é necessariamente própria.…”

Section: Introduc ¸ãOunclassified

Coloração Harmoniosa

Araújo

Martins

Santos

2022

Anais Do VII Encontro De Teoria Da Computação (ETC 2022)

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Uma k-coloração c: V(G)->{1, 2, ..., k} é uma coloração linha-distinguível se cada par de arestas distintas tem conjuntos de cores distintos em suas extremidades. O número cromático linha-distinguível de G é \lambda (G)=min {k\in\mathbb{N}: G tem uma k-coloração linha-distinguível}. Se uma coloração linha-distinguível c é própria, então c é uma coloração harmoniosa. O número cromático harmonioso, denotado por h(G), é o menor k\in \mathbb{N} tal que G tem uma k-coloração harmoniosa. Neste trabalho, apresentamos uma condição necessária e suficiente para h(G) ser k, para um grafo arbitrário e k\in \mathbb{N}. Após isso, apresentamos formulações de programação inteira para computar h(G) e alguns testes preliminares em grafos aleatórios com densidades de arestas distintas.

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An upper bound for the harmonious chromatic number of a graph

Abstract: An upper bound for the harmonious chromatic number of a graph G is given. Three corollaries of the theorem are theorems or improvements of the theorems of Miller and Pritikin.

Cited by 28 publications

References 1 publication

Approximate and exact results for the harmonious chromatic number

Approximate and exact results for the harmonious chromatic number

Graph coloring Parameters-A Survey

Coloração Harmoniosa

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