Importance: Medium ✭✭
Author(s): Noel, Jonathan A.
Recomm. for undergrads: yes
Posted by: Jon Noel
on: February 2nd, 2013
Conjecture   If $ G $ is a $ k $-chromatic graph on at most $ mk $ vertices, then $ \text{ch}(G)\leq \text{ch}(K_{m*k}) $.

For integers $ m,k\geq1 $, let $ K_{m*k} $ denote the complete $ k $-partite graph in which every part has size $ m $.

In one of the original papers on choosability, Erdos, Rubin and Taylor [ERT] proved that $ \text{ch}(K_{2*k})=k $. Later, Ohba [Ohba] conjectured the following generalization: if $ |V(G)|\leq 2\chi(G)+1 $, then TeX Embedding failed!.} This was proved by Noel, Reed and Wu [NRW12].

Theorem  (Noel, Reed and Wu 2012)   If $ |V(G)|\leq 2\chi(G)+1 $, then $ \text{ch}(G)=\chi(G) $.

The above theorem implies that the above conjecture holds for $ m=2 $. That is, if $ G $ is a $ k $-chromatic graph on at most $ 2k $ vertices (in fact, at most $ 2k+1 $ vertices), then $ \text{ch}(G)=k=\text{ch}(K_{2*k}) $.

Kierstead [Kie00] proved that $ \text{ch}(K_{3*k})=\left\lceil\frac{4k-1}{3}\right\rceil $. This was generalized by Noel, West, Wu and Zhu [NWWZ13] to the following:

Theorem  (Noel, West, Wu and Zhu 2013)   For every graph $ G $, \[\text{ch}(G)\leq\max\left\{\chi(G),\left\lceil\frac{|V(G)|+\chi(G)-1}{3}\right\rceil\right\}.\]

Therefore, if $ G $ is a $ k $-chromatic graph on at most $ 3k $ vertices, then $ \text{ch}(G)\leq \left\lceil\frac{4k-1}{3}\right\rceil=\text{ch}(K_{3*k}) $. This shows that the conjecture is true for $ m=3 $.

Recently, Kierstead, Salmon and Wang [KSW14] proved the following:

Theorem  (Kierstead, Salmon and Wang 2014)   $ \text{ch}(K_{4*k})=\left\lceil\frac{3k-1}{2}\right\rceil $.

However, it is not known whether the upper bound of $ \left\lceil\frac{3k-1}{2}\right\rceil $ holds for all $ k $-chromatic graphs on at most $ 4k $ vertices. If true, it would verify the conjecture for $ m=4 $.

The following is a refinement of the conjecture.

Conjecture  (Noel 2013)   For $ n\geq k\geq 1 $ there is a graph $ G_{n,k} $ such that
    \item $ G_{n,k} $ is a complete $ k $-partite graph on $ n $ vertices, \item the stability number of $ G_{n,k} $ is $ \left\lceil n/k\right\rceil $, and \item every $ k $-chromatic graph $ G $ on at most $ n $ vertices satisfies $ \text{ch}(G)\leq \text{ch}(G_{n,k}) $.


[Alo92] N. Alon. Choice numbers of graphs: a probabilistic approach. Combin. Probab. Comput., 1(2):107–114, 1992.

[ERT80] P. Erdos, A. L. Rubin, and H. Taylor. Choosability in graphs. Congress. Numer., XXVI, pages 125–157, 1980.

[Kie00] H. A. Kierstead. On the choosability of complete multipartite graphs with part size three. Discrete Math., 211(1-3):255–259, 2000.

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

*[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

[NWWZ13] J. A. Noel, D. B. West, H. Wu, and X. Zhu. Beyond Ohba's Conjecture: A bound on the choice number of $ k $-chromatic graphs with $ n $ vertices. Preprint, arXiv:1308.6739v1, August 2013. pdf

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

[Yan03] D. Yang. Extension of the game coloring number and some results on the choosability of complete multipartite graphs. PhD thesis, Arizona State University, Tempe, Arizona, 2003.

* indicates original appearance(s) of problem.


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