Equitable partition of plane graphs with independent crossings into induced forests

Niu, Bei; Zhang, Xin; Gao, Yuping

doi:10.1016/j.disc.2019.111792

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…In 2015, Esperet, Lemoine and Maffray [3] confirmed Conjecture 1.2 by showing that va ≡ (G) ≤ 4 for every planar graph G. Recently, Niu, Zhang and Gao [4] proved that va ≡ (G) ≤ 8 for every IC-planar graph G (a graph is IC-planar if it has embedding in the plane so that each edge is crossed by at most one other edge and each vertex is incident with at most one crossing edge).…”

Section: Introductionmentioning

confidence: 90%

Equitable partition of graphs into induced linear forests

Zhang

Niu

2019

J Comb Optim

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Section: Introductionmentioning

confidence: 90%

Equitable partition of graphs into induced linear forests

Zhang

Niu

2019

J Comb Optim

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“…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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“…-G is complete or bipartite [14]; -∆(G) ≥ (|G| − 1)/2 [16,18]; -∆(G) ≤ 3 [15]; -G is 5-degenerate [1]; -G is d-degenerate with ∆(G) ≥ 10d [17]; -G is IC-planar with ∆(G) ≥ 14 or g(G) ≥ 6 [12]; -G is a d-dimensional grid with d ∈ {2, 3, 4} [3].…”

Section: Introductionmentioning

confidence: 99%

Complexity of tree-coloring interval graphs equitably

Niu

Zhang

2020

Preprint

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An equitable tree-k-coloring of a graph is a vertex k-coloring such that each color class induces a forest and the size of any two color classes differ by at most one. In this work, we show that every interval graph G has an equitable tree-k-coloring for any integer k ≥ ⌈(∆(G) + 1)/2⌉, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree-k-coloring for a given integer k. For disjoint union of split graphs, or K1,r-free interval graphs with r ≥ 4, we prove that it is W [1]hard to decide whether there is an equitable tree-k-coloring when parameterized by number of colors, or by treewidth, number of colors and maximum degree, respectively.

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Equitable partition of plane graphs with independent crossings into induced forests

Cited by 8 publications

References 17 publications

Equitable partition of graphs into induced linear forests

Equitable partition of graphs into induced linear forests

On Equitable List Arboricity of Graphs

Complexity of tree-coloring interval graphs equitably

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