Choice number of complete multipartite graphs $K_{3*3,2*(k-5),1*2}$ and $K_{4,3*2,2*(k-6),1*3}$

TL;DR

This paper proves Ohba's conjecture for two new classes of complete multipartite graphs, confirming that their choice number equals their chromatic number, thus advancing understanding of graph coloring properties.

Contribution

The paper establishes that Ohba's conjecture holds for two specific classes of complete multipartite graphs, expanding the classes for which the conjecture is verified.

Findings

Ohba's conjecture is true for $K_{3*3,2*(k-5),1*2}$ for $k\\geq 5$.

Ohba's conjecture is true for $K_{4,3*2,2*(k-6),1*3}$ for $k\\geq 6$.

Confirmed the conjecture for new classes of graphs.

Abstract

A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=\chi(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2\chi(G)+1$ 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 $Ch(K_{3*3,2*(k-5),1*2})=k$ and $Ch(K_{4,3*2,2*(k-6),1*3})=k$ (for $k\geq 5$ and $k\geq 6$, respectively).