The relaxed game chromatic number of outerplanar graphs

Dunn, Charles; Kierstead, H. A.

doi:10.1002/jgt.10172

Cited by 13 publications

(7 citation statements)

References 15 publications

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“…This variation was recently studied by Chou et al in [2] and by the authors in [5,6]. In the usual version of the coloring game, two players, Alice and Bob, alternate coloring the uncolored vertices of a graph G with legal colors from a set of colors X ; with jX j ¼ r: A color aAX is legal for an uncolored vertex u , if u has no neighbors already colored with a: Alice wins this game if all of the vertices eventually are colored legally.…”

Section: Introductionmentioning

confidence: 88%

A simple competitive graph coloring algorithm III

Dunn

Kierstead

2004

Journal of Combinatorial Theory, Series B

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We consider the following game played on a finite graph G: Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G; using colors from a set X ; with jX j ¼ r: A color aAX is legal for an uncolored vertex v if by coloring v with a; the subgraph induced by all vertices of color a has maximum degree at most d: Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is planar, then Alice has a winning strategy for this game when r ¼ 3 and dX132: We also show that for sufficiently large d; if G is a planar graph without a 4-cycle or with girth at least 5; then Alice has a winning strategy for the game when r ¼ 2: r 2004 Elsevier Inc. All rights reserved.

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Section: Introductionmentioning

confidence: 88%

A simple competitive graph coloring algorithm III

Dunn

Kierstead

2004

Journal of Combinatorial Theory, Series B

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“…Let F be the class of forests, Q the class of outerplanar graphs, PK k the class of partial k-trees and P the class of planar graphs. The following theorems are combinations of results proved in a series of papers [3][4][5][6]8,10,11,16,18,19].…”

Section: Introductionmentioning

confidence: 91%

“…The relaxed game chromatic number was introduced in [3], and has been studied in [4][5][6]10,16]. For a class C of graphs, let…”

Section: Introductionmentioning

confidence: 99%

The 6-relaxed game chromatic number of outerplanar graphs

Zhu

2008

Discrete Mathematics

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Suppose G is a graph and k, d are integers. The (k, d)-relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Alice's goal is to have all the vertices coloured, and Bob's goal is the opposite: to have an uncoloured vertex without a legal colour. The d-relaxed game chromatic number of G, denoted by χ, is the least number k so that when playing the (k, d)-relaxed colouring game on G, Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then χ (d) g (G) ≤ 2 for d ≥ 6.

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“…For a fixed r, the r-game defect of G, denoted def g (G, r), is the least d such that Alice has a winning strategy. These parameters have been further examined in a number of papers, including [11,12,13,9,19].…”

Section: Introductionmentioning

confidence: 99%

The Relaxed Edge-Coloring Game and k-Degenerate Graphs

Dunn

Morawski

Nordstrom

2014

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Abstract. The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F ) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j).

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The relaxed game chromatic number of outerplanar graphs

Cited by 13 publications

References 15 publications

A simple competitive graph coloring algorithm III

A simple competitive graph coloring algorithm III

The 6-relaxed game chromatic number of outerplanar graphs

The Relaxed Edge-Coloring Game and k-Degenerate Graphs

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