Sum List Colorings of Wheels

Kemnitz, Arnfried; Marangio, Massimiliano; Voigt, Margit

doi:10.1007/s00373-015-1565-y

Cited by 9 publications

(3 citation statements)

References 6 publications

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“…It follows that starting with a θ-graph G and adding a G-path two times, in this special case (to obtain a generalized θ-graph), we always obtain a graph that is not sc-greedy. On the other hand, the graph presented in Figure 1 that needs to be added a G-path 3 times is still sc-greedy (see the graph G 10,12 in [10]). It leads to formulating the following problem.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Sum-list-colouring of θ-hypergraphs

2022

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Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Sum-list-colouring of θ-hypergraphs

2022

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“…Most of the graphs for which the sum choice number is known exactly are those which have been shown to be sc-greedy. These include complete graphs [9], trees [9], cycles [2], cycles with pendant paths [6], the Petersen graph [6], P 2 P n [6], generalised theta-graphs Θ k 1 ,k 2 ,k 3 (unless k 1 = k 2 = 1 and k 3 is odd), certain wheels [10], and trees of cycles [12,6].…”

Section: Background On the Sum Choice Numbermentioning

confidence: 99%

The interactive sum choice number of graphs

Bonamy

Meeks

2021

Discrete Applied Mathematics

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We introduce a variant of the well-studied sum choice number of graphs, which we call the interactive sum choice number. In this variant, we request colours to be added to the vertices' colour-lists one at a time, and so we are able to make use of information about the colours assigned so far to determine our future choices. The interactive sum choice number cannot exceed the sum choice number and we conjecture that, except in the case of complete graphs, the interactive sum choice number is always strictly smaller than the sum choice number. In this paper we provide evidence in support of this conjecture, demonstrating that it holds for a number of graph classes, and indeed that in many cases the difference between the two quantities grows as a linear function of the number of vertices.

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“…All sc-greedy graphs on five vertices were determined by Lastrina [16] and on six vertices by Kemnitz et al [13]. Moreover, all sc-greedy complete multipartite graphs [12], wheels [14], and broken wheels [8] were determined. Isaak [10] proved that block graphs are sc-greedy.…”

Section: Introductionmentioning

confidence: 99%

Sum-list colouring of unions of a hypercycle and a path with at most two vertices in common

Drgas-Burchardt¹,

Sidorowicz²

2020

Discuss. Math. Graph Theory

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Given a hypergraph H and a function f : V (H) −→ N, we say that H is f-choosable if there is a proper vertex colouring φ of H such that φ(v) ∈ L(v) for all v ∈ V (H), where L : V (H) −→ 2 N is any assignment of f (v) colours to a vertex v. The sum choice number Hi sc (H) of H is defined to be the minimum of v∈V (H) f (v) over all functions f such that H is f-choosable. For an arbitrary hypergraph H the inequality χ sc (H) ≤ |V (H)| + |E(H)| holds, and hypergraphs that attain this upper bound are called sc-greedy. In this paper we characterize sc-greedy hypergraphs that are unions of a hypercycle and a hyperpath having at most two vertices in common. Consequently, we characterize the hypergraphs of this type that are forbidden for the class of sc-greedy hypergraphs.

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Sum List Colorings of Wheels

Cited by 9 publications

References 6 publications

Sum-list-colouring of θ-hypergraphs

Sum-list-colouring of θ-hypergraphs

The interactive sum choice number of graphs

Sum-list colouring of unions of a hypercycle and a path with at most two vertices in common

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