On hamiltonian colorings for some graphs

Shen, Yufa; He, Wei; Li, Xue; He, Dongmei; Yang, Xiaojing

doi:10.1016/j.dam.2007.12.002

Cited by 4 publications

(16 citation statements)

References 5 publications

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“…We use an approach similar to the one used in [1] to derive a lower bound of the hamiltonian chromatic number of trees. We remark that our proof for the hamiltonian chromatic number of a special class of caterpillars is simple than one given in [7] by different approach. We also inform the readers that the hamiltonian chromatic number obtain in this paper is one less than that defined in [2,3,4,5,7] as we allowed 0 for coloring while they do not.…”

Section: Introductionmentioning

confidence: 79%

“…for a connected graph G of order n then such a graph G is called a graph with maximum distance bound n/2 or DB(n/2) graph for short. Shen et al [7] proved the following Theorems about DB(n/2) graphs and using it determined the hamiltonian chromatic number for double stars and a special class of caterpillars.…”

Section: Preliminariesmentioning

confidence: 99%

“…Theorem 2. ( [7]) Let G be a DB(n/2) graph of order n ≥ 4. Then for any σ, c σ is a hamiltonian coloring for G with hc(c σ ) = (n − 1) 2 + 1 − D(σ).…”

Section: Preliminariesmentioning

confidence: 99%

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On Hamiltonian Colorings of Trees

Bantva¹

2016

Algorithms and Discrete Applied Mathematics

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A hamiltonian coloring c of a graph G of order n is a mapping c : V (G) → {0, 1, 2, ...} such that D(u, v) + |c(u) − c(v)| ≥ n − 1, for every two distinct vertices u and v of G, where D(u, v) denotes the detour distance between u and v which is the length of a longest u, v-path in G. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of trees and give a sufficient condition to achieve this lower bound. Using this condition we determine the hamiltonian chromatic number of symmetric trees, firecracker trees and a special class of caterpillars.

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Section: Introductionmentioning

confidence: 79%

Section: Preliminariesmentioning

confidence: 99%

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On Hamiltonian Colorings of Trees

Bantva¹

2016

Algorithms and Discrete Applied Mathematics

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“…Note that any optimal hamiltonian coloring always assign label 0 to some vertex, then the span of any hamiltonian coloring c which is defined as max{|c(u)− c(v)| : u, v ∈ V (G)}, is the maximum integer used for coloring. However, in [7,8,10] only positive integers are used as colors. Therefore, the hamiltonian chromatic number defined in this article is one less than that defined in [7,8,10].…”

Section: Introductionmentioning

confidence: 99%

“…An upper bound for hc(P n ) was established by Chartrand et al in [7] but the exact value of hc(P n ) which is equal to the radio antipodal number ac(P n ) was determined by Khennoufa and Togni in [9]. In [10], Shen et al have discussed the hamiltonian chromatic number for graphs G with max{D(u, v) : u, v ∈ V (G), u = v} ≤ n/2, where n is the order of graph G and they gave the hamiltonian chromatic number for a special class of caterpillars and double stars. The hamiltonian chromatic number of block graphs and trees is discussed by Bantva in [1] and [2], respectively.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian chromatic number of block graphs

Bantva¹

2017

JGAA

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Let G be a simple connected graph of order n. A hamiltonian coloring c of a graph G is an assignment of colors (non-negative integers) to the vertices of G such that D(u, v) + |c(u) − c(v)| ≥ n − 1 for every two distinct vertices u and v of G, where D(u, v) denotes the detour distance between u and v in G which is the length of the longest path between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we give a necessary and sufficient condition to achieve a lower bound for the hamiltonian chromatic number of block graphs given in [1, Theorem 1]. We present an algorithm for optimal hamiltonian coloring of a special class of block graphs, namely SDB(p/2) block graphs. We characterize level-wise regular block graphs and extended star of blocks achieving this lower bound.

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On Hamiltonian Colorings of Block Graphs

Bantva¹

2016

WALCOM: Algorithms and Computation

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Abstract. A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that D(u, v) + |c(u)−c(v)| ≥ p − 1 for every two distinct vertices u and v of G, where D(u, v) denotes the detour distance between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of block graphs and give a sufficient condition to achieve the lower bound. We characterize symmetric block graphs achieving this lower bound. We present two algorithms for optimal hamiltonian coloring of symmetric block graphs.

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On hamiltonian colorings for some graphs

Cited by 4 publications

References 5 publications

On Hamiltonian Colorings of Trees

On Hamiltonian Colorings of Trees

Hamiltonian chromatic number of block graphs

On Hamiltonian Colorings of Block Graphs

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