Abstract
A graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely C h ( G ) = χ ( G ) . Ohba has conjectured that every graph G satisfying | V ( G ) | ≤ 2 χ ( G ) + 1 is chromatic-choosable. Since each k -chromatic graph is a subgraph of a complete k -partite graph, we see that Ohba’s conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba’s conjecture has been verified are: K 3 ∗ 2 , 2 ∗ ( k − 3 ) , 1 , K 3 , 2 ∗ ( k − 1 ) , K s + 3 , 2 ∗ ( k − s − 1 ) , 1 ∗ s , K 4 , 3 , 2 ∗ ( k − 4 ) , 1 ∗ 2 , and K 5 , 3 , 2 ∗ ( k − 5 ) , 1 ∗ 3 . In this paper, we show that Ohba’s conjecture is true for two new classes of complete multipartite graphs: graphs with three parts of size 3 and graphs with one part of size 4 and two parts of size 3. Namely, we prove that C h ( K 3 ∗ 3 , 2 ∗ ( k − 5 ) , 1 ∗ 2 ) = k and C h ( K 4 , 3 ∗ 2 , 2 ∗ ( k − 6 ) , 1 ∗ 3 ) = k (for k ≥ 5 and k ≥ 6 , respectively).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
