Hamiltonian colorings of graphs

Chartrand, Gary; Nebeský, Ladislav; Zhang, Ping

doi:10.1016/j.dam.2004.08.007

Cited by 12 publications

(43 citation statements)

References 2 publications

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“…Moreover, equality holds in (2) if u and v are in different branches when ω = 1 and in opposite branches when ω ≥ 2.…”

Section: A Lower Bound For Hamiltonian Chromatic Number Of Block Graphsmentioning

confidence: 99%

“…Thus the hamiltonian chromatic number of a connected graph G measures how close G is to being hamiltonian-connected, minimum the hamiltonian chromatic number of a connected graph G is, the closer G is to being hamiltonian-connected. The concept of hamiltonian coloring was introduced by Chartrand et al [2] as a variation of radio k-coloring of graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Without loss of generality, we initiate with label 0, 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 [2,3,6] only positive integers are used as colors. Therefore, the hamiltonian chromatic number defined in this article is one less than that defined in [2,3,6] and hence we will make necessary adjustment when we present the results of [2,3,6] in this article.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Moreover, equality holds in (2) if u and v are in different branches when ω = 1 and in opposite branches when ω ≥ 2.…”

Section: A Lower Bound For Hamiltonian Chromatic Number Of Block Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Hamiltonian Colorings of Block Graphs

Bantva¹

2016

WALCOM: Algorithms and Computation

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“…Chartrand et al [7] observed that in case of paths P n , d = n − 1 and d(u, v) is same as the the length of a longest u − v path which is denoted by D(u, v) known as detour distance between u and v, then (1) is equivalent to D(u, v) + |c(u) − c(v)| ≥ n − 1.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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 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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Hamiltonian colorings of graphs

Cited by 12 publications

References 2 publications

On Hamiltonian Colorings of Block Graphs

On Hamiltonian Colorings of Block Graphs

Hamiltonian chromatic number of block graphs

On Hamiltonian Colorings of Trees

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