Equitable coloring and equitable choosability of graphs with small maximum average degree

Dong, Aijun; Zhang, Xin

doi:10.7151/dmgt.2049

Cited by 7 publications

(7 citation statements)

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“…In [12] 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 [24], series-parallel graphs [22], graphs with small maximum average degree [3], powers of cycles [9], and certain planar graphs (see [2], [13], [23], and [25]). In 2013, Kierstead and Kostochka made substantial progress on Conjecture 3, and proved it for all graphs of maximum degree at most 7 (see [11]).…”

Section: Conjecture 3 ([12]) Every Graph G Is Equitablymentioning

confidence: 80%

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A Note on the Equitable Choosability of Complete Bipartite Graphs

Mudrock¹,

Chase²,

Kadera³

et al. 2018

Preprint

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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 equitableIn this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West impliesWe 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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Section: Conjecture 3 ([12]) Every Graph G Is Equitablymentioning

confidence: 80%

“…Conjecture 2 has been proven true for interval graphs, bipartite graphs, outerplanar graphs, subcubic graphs, certain planar graphs, and several other classes of graphs (see [1], [2], [3], [14], [15] and [21]).…”

Section: Theorem 1 ([6]mentioning

confidence: 99%

A Note on the Equitable Choosability of Complete Bipartite Graphs

Mudrock¹,

Chase²,

Kadera³

et al. 2018

Preprint

View full text Add to dashboard Cite

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 equitableIn this note we study the equitable choosability of complete bipartite graphs. A result of Kostochka, Pelsmajer, and West impliesWe 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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“…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%

Section: Conjecture 2 ([1] ∆-Ecc)mentioning

confidence: 99%

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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“…Their conjecture is still open and is known as the ∆-Equitable Coloring Conjecture (∆-ECC for short). It has received considerable attention in the literature (see e.g., [5,8,9,17,18,27]).…”

Section: Equitable Coloring and Equitable Choosabilitymentioning

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 coloring and equitable choosability of graphs with small maximum average degree

Cited by 7 publications

References 0 publications

A Note on the Equitable Choosability of Complete Bipartite Graphs

A Note on the Equitable Choosability of Complete Bipartite Graphs

A note on the equitable choosability of complete bipartite graphs

On Equitable List Arboricity of Graphs

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