A Proof of a Conjecture of Ohba
Jonathan A. Noel, Bruce A. Reed and Hehui Wu
A list colouring problem asks the following: given an assignment of lists
L(v) of colours to each vertex v of a graph G, does there exist a proper
colouring f of G such that f (v) ∈ L(v) for every vertex v? Such a colouring
is called an acceptable colouring for L. The list chromatic number of G,
denoted χ` (G), is defined to be the minimum k such that G has an acceptable
L-colouring whenever |L(v)| ≥ k for all v ∈ V (G). Of course, a proper kcolouring of G is equivalent to an acceptable colouring for L where L(v) =
{1, 2, . . . , k} for all v ∈ V (G). Therefore, we have the following bound:
χ ≤ χ` .
List colouring was introduced independently by Vizing [7] and Erdős et
al. [1] and has become a popular topic among researchers (see e.g. [8]). Of
particular interest is the problem of determining classes of graphs which
satisfy χ` = χ. These graphs are said to be chromatic-choosable [3]. In this
paper, we prove the following conjecture of Ohba [3].
Conjecture 1 (Ohba [3]). If |V (G)| ≤ 2χ(G) + 1, then G is chromaticchoosable.
It is straightforward to see that adding edges between vertices in different colour classes in a χ(G)-colouring does not decrease χ` and so Ohba’s
Conjecture is true for all graphs if and only if it is true for complete multipartite graphs. Thus, let G be a complete k-partite graph on at most 2k + 1
vertices.
Ohba’s Conjecture has received considerable attention and many partial
results have been proven. Reed and Sudakov proved that G is chromaticchoosable if |V (G)| ≤ 35 k − 43 [5] or if |V (G)| ≤ (2 − o(1))k [4]. He et al. [6]
proved Ohba’s Conjecture for graphs of stability number at most 3; the case
for stability number at most 5 was proven by Kostochka et al. [2].
Our proof is divided into three parts based on the following lemmas.
Before stating these lemmas, we require some definitions.
Definition 2. Say that a colour c ∈ ∪v∈V (G) L(v) is frequent if it appears
in the lists of at least k + 1 vertices of G.
Definition 3. A proper colouring f : V (G) → ∪v∈V (G) L(v) is said to be a
near-acceptable colouring for L if for every v ∈ V (G) either
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• f (v) ∈ L(v), or
• f (v) is frequent and f −1 (f (v)) = {v}.
Lemma 4. If there is a near-acceptable colouring for L, then there is an
acceptable colouring for L.
Lemma 5. If there are at least k frequent colours, then there is a nearacceptable colouring for L.
Lemma 6. If Ohba’s Conjecture is false, then there is a counterexample
with at least k frequent colours.
Clearly these three lemmas are enough to prove Ohba’s Conjecture. To
prove them, we combine Hall’s Theorem, greedy colouring procedures and
several counting arguments.
References
[1] P. Erdős, A. L. Rubin, and H. Taylor, Choosability in graphs, Proceedings
of the West Coast Conference on Combinatorics, Graph Theory and
Computing, Congress. Numer., XXVI, 1980, pp. 125–157.
[2] A. V. Kostochka, M. Stiebitz, and D. R. Woodall, Ohba’s conjecture
for graphs with independence number five, Discrete Math. 311 (2011),
no. 12, 996–1005.
[3] K. Ohba, On chromatic-choosable graphs, J. Graph Theory 40 (2002),
no. 2, 130–135.
[4] B. Reed and B. Sudakov, List colouring of graphs with at most (2−o(1))χ
vertices, Proceedings of the International Congress of Mathematicians,
Vol. III (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 587–603.
[5]
, List colouring when the chromatic number is close to the order
of the graph, Combinatorica 25 (2005), no. 1, 117–123.
[6] Y. Shen, W. He, G. Zheng, and Y. Li, Ohba’s conjecture is true for
graphs with independence number at most three, Appl. Math. Lett. 22
(2009), no. 6, 938–942.
[7] V. G. Vizing, Coloring the vertices of a graph in prescribed colors,
Diskret. Analiz (1976), no. 29 Metody Diskret. Anal. v Teorii Kodov
i Shem, 3–10, 101.
[8] D. R. Woodall, List colourings of graphs, Surveys in combinatorics, 2001
(Sussex), London Math. Soc. Lecture Note Ser., vol. 288, Cambridge
Univ. Press, Cambridge, 2001, pp. 269–301.
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