On-Line Ohba's Conjecture

Recomm. for undergrads: yes
Posted by: Jon Noel
on: April 11th, 2013
Conjecture   If $ |V(G)|\leq 2\chi(G) $, then the on-line choice number of $ G $ is equal to $ \chi(G) $.

The on-line choice number of a graph $ G $ is determined by the following $ 2 $-player game. We call the players Painter and Lister. Let $ C=\{c_1,c_2,\dots\} $ be an infinite set of colours.

Initially, every vertex $ v $ of $ G $ is given a list $ L(v):=\emptyset $ of colours. At the $ i $th step of the game, Lister chooses a non-empty set $ V_i $ of vertices which are not coloured and, for each vertex $ v\in V_i $, the list of $ v $ is changed to $ L(v)\cup\{c_i\} $. Then, painter chooses a stable subset of $ V_i $ to colour with $ c_i $.

If, at some step of the game, there is a vertex $ v\in V_i $ such that $ |L(v)|=k $ and $ v $ is not coloured with $ c_i $, then Lister wins the game. Otherwise, if every vertex is eventually coloured, then Painter wins. The on-line choice number of $ G $, denoted $ \text{ch}^{\text{OL}}(G) $ is defined to be the minimum $ k $ such that Painter has a winning strategy.

The on-line choice number was introduced independently by Schauz [Sch09] and Zhu [Zhu09].

It is clear that the on-line choice number is at least as large as the choice number; ie. $ \text{ch}^{\text{OL}}(G)\geq \text{ch}(G) $. That is, if there is an assignment $ L $ of lists of size $ k $ such that there is no acceptable colouring, then Lister can simply play so that the resulting lists are given by $ L $.

For choosability, Ohba [Ohb02] made the following conjecture, which was proved by Noel, Reed and Wu [NRW12]: if $ |V(G)|\leq 2\chi(G)+1 $, then TeX Embedding failed!.}

The On-Line Ohba's Conjecture, proposed by Huang, Wong and Zhu [HWZ12], assumes a stronger bound on the number of vertices (namely, $ |V(G)|\leq 2\chi(G) $). A result of Kim, Kwon, Liu and Zhu [KKLZ12] is that, for $ k\geq 3 $, the complete $ k $-partite graph with one part of size $ 3 $ and $ k-1 $ parts of size $ 2 $ has on-line choice number $ k+1 $. Therefore, On-Line Ohba's Conjecture cannot be extended to graphs on $  2\chi(G)+1 $ vertices.

In [KSW14], it is proved that the complete $ 3 $-partite graph with parts of size $ 4 $ has choice number $ 4 $, but on-line choice number at least $ 5 $. It is also claimed that a computer check shows that $ 5 $ is the correct value for the on-line choice number (no formal proof is given).

Some partial results are known:

Theorem  (Huang et al. 2012)   The on-line choice number of the complete $ k $-partite graph with all parts of size $ 2 $ is $ k $.
Theorem  (Kozik, Micek, Zhu 2012)   On-Line Ohba's Conjecture is true for graphs of independence number at most $ 3 $.
Theorem  (Kozik et al. 2012)   If $ |V(G)|\leq \chi(G)+\sqrt{\chi(G)} $, then $ \text{ch}^{\text{OL}}(G)=\chi(G) $.
Theorem  (Carraher et al. 2012)   If $ |V(G)|\leq \chi(G)+2\sqrt{\chi(G)-1} $, then $ \text{ch}^{\text{OL}}(G)=\chi(G) $.

Presently, it is not known whether there is a real constant $ a>1 $ such that if $ |V(G)|\leq a\chi(G) $, then $ \text{ch}^{\text{OL}}(G)=\chi(G) $.


[CLM+13] J. Carraher, S. Loeb, T. Mahoney, G. Puleo, M.-T. Tsai, and D. West. Three Topics in Online List Coloring. Preprint, February 2013.

*[HWZ12] P. Huang, T. Wong, and X. Zhu. Application of polynomial method to on-line list colouring of graphs. European J. Combin., 33(5):872–883, 2012.

[KSW14] H. A. Kierstead, A. Salmon and R. Wang. On the Choice Number of Complete Multipartite Graphs With Part Size Four.

[KKLZ12] S.-J. Kim, Y. S. Kwon, D. D.-F. Liu, and X. Zhu. On-line list colouring of complete multipartite graphs. Electron. J. Combin., 19(1):Paper 41, 13, 2012.

[KMZ12] J. Kozik, P. Micek, and X. Zhu. Towards on-line Ohba’s conjecture. Preprint, arXiv:1111.5458v2, December 2012.

[Noe13] J. A. Noel. Choosability of Graphs With Bounded Order: Ohba's Conjecture and Beyond. Master's thesis, McGill University, Montreal. pdf

[NRW12] J. A. Noel, B. A. Reed, and H. Wu. A Proof of a Conjecture of Ohba. Preprint, arXiv:1211.1999v1, November 2012. Webpage

[Ohb02] K. Ohba. On chromatic-choosable graphs. J. Graph Theory, 40(2):130–135, 2002.

[Sch09] U. Schauz. Mr. Paint and Mrs. Correct. Electron. J. Combin., 16(1):Research Paper 77, 18, 2009.

[Sch10] U. Schauz. A paintability version of the combinatorial Nullstellensatz, and list colorings of k-partite k-uniform hypergraphs. Electron. J. Combin., 17(1):Research Paper 176, 13, 2010.

[Zhu09] X. Zhu. On-line list colouring of graphs. Electron. J. Combin., 16(1):Research Paper 127, 16, 2009.

* indicates original appearance(s) of problem.