Equitable list vertex colourability and arboricity of grids

Drgas-Burchardt, Ewa; Dybizbański, Janusz; Furmańczyk, Hanna; Sidorowicz, Elżbieta

doi:10.2298/fil1818353d

Cited by 10 publications

(10 citation statements)

References 13 publications

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“…The first statement of Theorem 1 generalizes the result obtained in [3] for d ∈ {1, 2}. In this paper we present an algorithm that confirms both, the first and second statements of Theorem 1 for all possible d. The algorithm, given in Sect.…”

Section: Motivation and Preliminariessupporting

confidence: 73%

Equitable d-degenerate Choosability of Graphs

Drgas-Burchardt

Furmańczyk

Sidorowicz

2020

Lecture Notes in Computer Science

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Let D d be the class of d-degenerate graphs and let L be a list assignment for a graph G. A colouring of G such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a d-degenerate graph is called the (L,colouring is a generalization of an equitable list coloring, introduced by Kostochka et al., and an equitable list arboricity presented by Zhang. Such a model can be useful in the network decomposition where some structural properties on subnets are imposed. In this paper we give a polynomial-time algorithm that for a given (k, d)-partition of G with a t-uniform list assignment L and t ≥ k, returns its equitable (L, D d−1 )colouring. In addition, we show that 3-dimensional grids are equitably (L, D1)-colorable for any t-uniform list assignment L where t ≥ 3.

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Section: Motivation and Preliminariessupporting

confidence: 73%

Equitable d-degenerate Choosability of Graphs

Drgas-Burchardt

Furmańczyk

Sidorowicz

2020

Lecture Notes in Computer Science

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“…In [13] it is shown that Conjectures 3 and 4 hold for forests, complete bipartite graphs, connected interval graphs, and 2-degenerate graphs with maximum degree at least 5. Conjectures 3 and 4 have also been verified for outerplanar graphs [25], series-parallel graphs [23], graphs with small maximum average degree [3], certain graphs related to grids [4], powers of cycles [10], and certain planar graphs (see [2,14,24] and [26]). In 2013, Kierstead and Kostochka made substantial progress on Conjecture 3, and proved it for all graphs of maximum degree at most 7 (see [12]).…”

Section: Equitable Choosabilitymentioning

confidence: 90%

A note on the equitable choosability of complete bipartite graphs

Chase¹,

Kadera²,

Mudrock³

et al. 2021

Discuss. Math. Graph Theory

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In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A k-assignment, L, for a graph G assigns a list, L(v), of k available colors to each v ∈ V (G), and an equitable L-coloring of G is a proper coloring, f , of G such that f (v) ∈ L(v) for each v ∈ V (G) and each color class of f has size at most ⌈|V (G)|/k⌉. Graph G is said to be equitably k-choosable if an equitable L-coloring of G exists whenever L is a k-assignment for G. In this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies K n,m is equitably k-choosable if k ≥ max{n, m} provided K n,m = K 2l+1,2l+1. We prove K n,m is equitably k-choosable if m ≤ ⌈(m + n)/k⌉ (k − n) which gives K n,m is equitably k-choosable for certain k satisfying k < max{n, m}. We also give a complete characterization of the equitable choosability of complete bipartite graphs that have a partite set of size at most 2.

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“…We verify this stronger version of Zhang's Conjecture for powers of paths, 2-degenerate graphs, and certain other graphs. We also show that if G is equitably k-list arborable it does not necessarily follow that G is equitably (k + 1)-list arborable which addresses a question of Drgas-Burchardt, Furmańczyk, and Sidorowicz (2018).…”

mentioning

confidence: 84%

“…Indeed, K 9,9 is equitably vertex 2-arborable, but it is not equitably vertex 3-arborable. The following conjecture of Wu, Zhang, and Li is well-known and has received some attention in the literature (see e.g., [6,23,26,29,30,31,32]).…”

Section: List Vertex Arboricity and Equitable Vertex Arboricitymentioning

confidence: 99%

On Equitable List Arboricity of Graphs

Kaul¹,

Mudrock²,

Pelsmajer³

2020

Preprint

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Equitable list arboricity, introduced by Zhang in 2016, generalizes the notion of equitable list coloring by requiring each color class to be acyclic (instead of an independent set) in addition to the usual upper bound on the size of each color class. Graph G is equitably k-list arborable if an equitable, arborable list coloring of G exists for every list assignment for G that associates with each vertex in G a list of k available colors. Zhang conjectured that any graph G is equitably k-list arborable for each k satisfying k ≥ ⌈(1 + ∆(G))/2⌉. We verify this conjecture for powers of cycles by applying a new lemma which is a general tool for extending partial equitable, arborable list colorings. We also propose a stronger version of Zhang's Conjecture for certain connected graphs: any connected graph G is equitably k-list arborable for each k satisfying k ≥ ⌈∆(G)/2⌉ provided G is neither a cycle nor a complete graph of odd order. We verify this stronger version of Zhang's Conjecture for powers of paths, 2-degenerate graphs, and certain other graphs. We also show that if G is equitably k-list arborable it does not necessarily follow that G is equitably (k + 1)-list arborable which addresses a question of Drgas-Burchardt, Furmańczyk, and Sidorowicz (2018).

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Equitable list vertex colourability and arboricity of grids

Cited by 10 publications

References 13 publications

Equitable d-degenerate Choosability of Graphs

Equitable d-degenerate Choosability of Graphs

A note on the equitable choosability of complete bipartite graphs

On Equitable List Arboricity of Graphs

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