Equitable and equitable list colorings of graphs

Zhu, Junlei; Bu, Yuehua

doi:10.1016/j.tcs.2010.06.027

Cited by 15 publications

(9 citation statements)

References 8 publications

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“…We say that a graph G is equitably k-colorable if In [11] it is shown that Conjectures 3 and 4 hold for forests, connected interval graphs, and 2-degenerate graphs with maximum degree at least 5. Conjectures 3 and 4 have also been verified for outerplanar graphs [29], series-parallel graphs [27], and certain planar graphs (see [12], [28], and [30]). In 2013, Kierstead and Kostochka made substantial progress on Conjecture 3, and proved it for all graphs of maximum degree at most 7 (see [10]).…”

Section: Equitable Coloringmentioning

confidence: 78%

Proportional choosability: A new list analogue of equitable coloring

Kaul

Mudrock

Pelsmajer

et al. 2019

Discrete Mathematics

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In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A k-assignment L for a graph G specifies a list L(v) of k available colors for each vertex v of G. An L-coloring assigns a color to each vertex v from its list L(v). For each color c, let η(c) be the number of vertices v whose listWe show that if a graph G is proportionally k-choosable, then every subgraph of G is also proportionally k-choosable and also G is proportionally (k + 1)-choosable, unlike equitable choosability for which analogous claims would be false. We also show that any graph G is proportionally k-choosable whenever k ≥ ∆(G) + ⌈|V (G)|/2⌉, and we use matching theory to completely characterize the proportional choosability of stars and the disjoint union of cliques.

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Section: Equitable Coloringmentioning

confidence: 78%

Proportional choosability: A new list analogue of equitable coloring

Kaul

Mudrock

Pelsmajer

et al. 2019

Discrete Mathematics

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show abstract

“…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%

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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“…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: 89%

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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Equitable and equitable list colorings of graphs

Cited by 15 publications

References 8 publications

Proportional choosability: A new list analogue of equitable coloring

Proportional choosability: A new list analogue of equitable coloring

A Note on the Equitable Choosability of Complete Bipartite Graphs

A note on the equitable choosability of complete bipartite graphs

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