# What are some ways to prove that a k-partite graph is nonplanar?

I am reading papers on graph theory and I encountered one work that talked about the planarity of a certain graph. In one of the proofs of a theorem, the author stated that the graph $$K_{1,2,3}$$ is nonplanar. Is this a known result? How can one check that this is planar? And does it hold for other graphs of the form $$K_{1,2,n}$$?

• What is $K_{1,2,3}$? Can you also reference the paper? I only know that $K_5$ and $K_{3,3}$ are non-planar. Commented Jul 26, 2020 at 5:35
• I may be wrong, but doesn't $K_{1,2,n}$ include $K_{1,2,3}$ as a subrgaph for $n\ge 3$? Therefore can't be planar if it's true that $K_{1,2,3}$ is not. Also see wiki/Euler characteristic. Commented Jul 26, 2020 at 5:35
• @Dmitry: It's the complete tripartite graph whose parts have cardinalities $1,2$, and $3$. In effect it's $K_{3,3}$ with two extra edges. Commented Jul 26, 2020 at 5:40

$$K_{1,2,3}$$ contains a copy of $$K_{3,3}$$, which is well known not to be planar.
• @Brenda: Almost: a subgraph that is a subdivision of $K_5$ or $K_{3,3}$. Commented Jul 26, 2020 at 6:03