Total Equitable List Coloring

Kaul, Hemanshu; Mudrock, Jeffrey A.; Pelsmajer, Michael J.

doi:10.1007/s00373-018-1965-x

Cited by 7 publications

(11 citation statements)

References 35 publications

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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: 79%

“…It is also shown that this bound is tight for forests. Also, in [9], it is conjectured that if T is a total graph, then T is equitably k-choosable for each k ≥ max{χ ℓ (T ), ∆(T )/2+2} where χ ℓ (T ) is the smallest m such that T is m-choosable. In this note we will present some results on the equitable choosability of complete bipartite graphs that will give us equitable k-choosability for values of k that are smaller than the maximum degree of the graph.…”

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

confidence: 99%

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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: 79%

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

confidence: 99%

A Note on the Equitable Choosability of Complete Bipartite Graphs

Mudrock¹,

Chase²,

Kadera³

et al. 2018

Preprint

Self Cite

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%

“…It is also shown that this bound is tight for forests. Also, in [10], it is conjectured that if T is a total graph, then T is equitably k-choosable for each k ≥ max{χ ℓ (T ), ∆(T )/2 + 2} where χ ℓ (T ) is the smallest m such that T is m-choosable. In this note we will present some results on the equitable choosability of complete bipartite graphs that will give us equitable k-choosability for values of k that are smaller than the maximum degree of the graph.…”

Section: Equitable Choosabilitymentioning

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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“…It is also shown that this bound is tight for forests. Also, in [8], it is conjectured that if T is a total graph, then T is equitably k-choosable for each k ≥ max{χ ℓ (T ), ∆(T )/2+2} where χ ℓ (T ), the list chromatic number of T , is the smallest m such that T is m-choosable. Finally, in [9], it is remarked that determining precisely which graphs are equitably 2-choosable is open.…”

Section: Equitable Choosabilitymentioning

confidence: 99%

On the Equitable Choosability of the Disjoint Union of Stars

Kaul¹,

Mudrock²,

Wagstrom³

2020

Preprint

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Equitable k-choosability is a list analogue of equitable k-coloring that was introduced by Kostochka, Pelsmajer, and West in 2003. It is known that if vertex disjoint graphs G 1 and G 2 are equitably k-choosable, the disjoint union of G 1 and G 2 may not be equitably k-choosable. Given any m ∈ N the values of k for which K 1,m is equitably k-choosable are known. Also, a complete characterization of equitably 2-choosable graphs is not known. With these facts in mind, we study the equitable choosability of n i=1 K 1,mi , the disjoint union of n stars. We show that determining whether n i=1 K 1,mi is equitably choosable is NP-complete when the same list of two colors is assigned to every vertex. We completely determine when the disjoint union of two stars (or n ≥ 2 identical stars) is equitably 2-choosable, and we present results on the equitable k-choosability of the disjoint union two stars for arbitrary k.

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Total Equitable List Coloring

Abstract: An equitable coloring is a proper coloring of a graph such that the sizes of the color classes differ by at most one. A graph G is equitably k-colorable if there exists an equitable coloring of G which uses k colors, each one appearing on either ⌊|V (

Cited by 7 publications

References 35 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 the Equitable Choosability of the Disjoint Union of Stars

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