# Choice-Perfect Graphs

• Volume: 33, Issue: 1, page 231-242
• ISSN: 2083-5892

top

## Abstract

top
Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring ϕ : V → S v2V Lv such that ϕ(v) ∈ Lv for all v ∈ V and ϕ(u) 6= ϕ(v) for all uv ∈ E. If such a ϕ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list contains at least k colors. In this note we initiate the study of graphs in which the choice number equals the clique number or the chromatic number in every induced subgraph. We call them choice-ω-perfect and choice-χ-perfect graphs, respectively. The main result of the paper states that the square of every cycle is choice-χ-perfect.

## How to cite

top

Zsolt Tuza. "Choice-Perfect Graphs." Discussiones Mathematicae Graph Theory 33.1 (2013): 231-242. <http://eudml.org/doc/267681>.

@article{ZsoltTuza2013,
abstract = {Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring ϕ : V → S v2V Lv such that ϕ(v) ∈ Lv for all v ∈ V and ϕ(u) 6= ϕ(v) for all uv ∈ E. If such a ϕ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list contains at least k colors. In this note we initiate the study of graphs in which the choice number equals the clique number or the chromatic number in every induced subgraph. We call them choice-ω-perfect and choice-χ-perfect graphs, respectively. The main result of the paper states that the square of every cycle is choice-χ-perfect.},
author = {Zsolt Tuza},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph coloring; list coloring; choice-perfect graph},
language = {eng},
number = {1},
pages = {231-242},
title = {Choice-Perfect Graphs},
url = {http://eudml.org/doc/267681},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Zsolt Tuza
TI - Choice-Perfect Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 1
SP - 231
EP - 242
AB - Given a graph G = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring ϕ : V → S v2V Lv such that ϕ(v) ∈ Lv for all v ∈ V and ϕ(u) 6= ϕ(v) for all uv ∈ E. If such a ϕ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list contains at least k colors. In this note we initiate the study of graphs in which the choice number equals the clique number or the chromatic number in every induced subgraph. We call them choice-ω-perfect and choice-χ-perfect graphs, respectively. The main result of the paper states that the square of every cycle is choice-χ-perfect.
LA - eng
KW - graph coloring; list coloring; choice-perfect graph
UR - http://eudml.org/doc/267681
ER -

## References

top
1. [1] N. Alon and M. Tarsi, Colorings and orientations of graphs, Combinatorica 12 (1992) 125-134. doi:10.1007/BF01204715[Crossref] Zbl0756.05049
2. [2] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50. doi:10.7151/dmgt.1037[Crossref] Zbl0902.05026
3. [3] M. Borowiecki, E. Sidorowicz and Zs. Tuza, Game list colouring of graphs, Electron. J. Combin. 14 (2007) #R26.
4. [4] P. Erdős, A.L. Rubin and H. Taylor, Choosability in graphs, West-Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, California, Congr. Numer. XXVI (1979) 125-157.
5. [5] H. Fleischner and M. Stiebitz, A solution to a colouring problem of P. Erdős, Discrete Math. 101 (1992) 39-48. doi:10.1016/0012-365X(92)90588-7[Crossref]
6. [6] F. Galvin, The list chromatic index of a bipartite multigraph, J. Combin. Theory (B) 63 (1995) 153-158. doi:10.1006/jctb.1995.1011[Crossref] Zbl0826.05026
7. [7] S. Gravier and F. Maffray, Graphs whose choice number is equal to their chromatic number, J. Graph Theory 27 (1998) 87-97. doi:10.1002/(SICI)1097-0118(199802)27:2h87::AID-JGT4i3.0.CO;2-B[Crossref] Zbl0891.05031
8. [8] S. Gravier and F. Maffray, On the choice number of claw-free perfect graphs, Discrete Math. 276 (2004) 211-218. doi:10.1016/S0012-365X(03)00292-9[Crossref] Zbl1031.05055
9. [9] A.J.W. Hilton and P.D. Johnson, Jr., Extending Hall’s theorem, in: Topics in Combinatorics and Graph Theory-Essays in Honour of Gerhard Ringel (R. Bodendiek et al., Eds.), (Teubner, 1990) 359-371.
10. [10] A.J.W. Hilton and P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999) 227-245. doi:10.1016/S0166-218X(99)00023-2[Crossref] Zbl0934.05001
11. [11] M. Juvan, B. Mohar and R. ˇSkrekovski, List total colourings of graphs, Combin. Probab. Comput. 7 (1998) 181-188. doi:10.1017/S0963548397003210[Crossref]
12. [12] M. Juvan, B. Mohar and R. Thomas, List edge-colorings of series-parallel graphs, Electron. J. Combin. 6 (1999) #R42. Zbl0939.05039
13. [13] D. Peterson and D.R. Woodall, Edge-choosability in line-perfect multigraphs, Discrete Math. 202 (1999) 191-199. doi:10.1016/S0012-365X(98)00293-3[Crossref] Zbl0928.05017
14. [14] Zs. Tuza, Graph colorings with local constraints-A survey, Discuss. Math. Graph Theory 17 (1997) 161-228. doi:10.7151/dmgt.1049[Crossref] Zbl0923.05027
15. [15] Zs. Tuza, Choice-perfect graphs and Hall numbers, manuscript, 1997.
16. [16] Zs. Tuza, Extremal jumps of the Hall number, Electron. Notes Discrete Math. 28 (2007) 83-89. doi:10.1016/j.endm.2007.01.012[Crossref]
17. [17] Zs. Tuza, Hall number for list colorings of graphs: Extremal results, Discrete Math. 310 (2010) 461-470. doi:10.1016/j.disc.2009.03.025[WoS][Crossref]
18. [18] Zs. Tuza and M. Voigt, List colorings and reducibility, Discrete Appl. Math. 79 (1997) 247-256. doi:10.1016/S0166-218X(97)00046-2[Crossref]
19. [19] V.G. Vizing, Coloring the vertices of a graph in prescribed colors, Metody Diskret. Anal. v Teorii Kodov i Schem 29 (1976) 3-10 (in Russian).
20. [20] D.R. Woodall, Edge-choosability of multicircuits, Discrete Math. 202 (1999) 271-277. doi:10.1016/S0012-365X(98)00297-0[Crossref] Zbl0928.05018

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.