Importance: Low ✭
Author(s): Melnikov, L. S.
Recomm. for undergrads: no
Posted by: asp
on: March 3rd, 2013
Problem   The valency-variety $ w(G) $ of a graph $ G $ is the number of different degrees in $ G $. Is the chromatic number of any graph $ G $ with at least two vertices greater than $$\ceil{ \frac{\floor{w(G)/2}}{|V(G)| - w(G)} } ~ ?$$

According to Jensen and Toft [JT, p. 90], the problem is due to Melnikov and was mentioned by Vizing [V] and Zykov [Z]. According to Zykov [Z], Melnikov showed that the suggested lower bound would be best possible.

A best possible upper bound on the chromatic number in terms of $ |V(G)| $ and $ w(G) $ is $ |V(G)| - \floor{ w(G) / 2 } $ as proved by Nettleton [N] and Dirac [D].


[D] G. A. Dirac. Valency-variey and chromatic number of abstract graphs. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 13, 59--64, 1964.

[JT] Tommy R. Jensen, Bjarne Toft: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Inc., New York, 1995.

[N] R. E. Nettleton. Some generalized theorems on connectivity. Canad. J. Math. 12, 546--554, 1960.

*[V] V. G. Vizing. Some unsolved problems in graph theory (in Russian). Uspekhi Mat. Nauk. 23, 117--134, 1968. English translation in Russian Math. Surveys 23, 125--141.

*[Z] A. A. Zykov. Problem 11. In: H. Sachs, H.-J. Voss, and H. Walther, editors, Beiträge zur Graphentheorie vorgetragen auf dem Internationalen Kolloquium in Manebach DDR vom 9.-12. Mai 1967, page 228. B. G. Teubner, 1968.

* indicates original appearance(s) of problem.


Comments are limited to a maximum of 1000 characters.
More information about formatting options