Let $G$ be a planar graph, which we may assume to be a triangulation, with vertex set $V$ and edge set $E$. Suppose the minimum vertex degree is at least 3, and suppose any two distinct edges share at most one vertex.

**Definition.** Given an edge $e \in E$ with vertices $v_1$ and $v_2$, define $D(e) = \mathrm{deg}(v_1). \mathrm{deg}(v_2)$, the product of the degrees of the vertices of $e$.

I have been calling this quantity the *density* of the edge $e$, but:

**Question 1:** Has this quantity $D(e)$ been previously studied? Does it already have a name?

This quantity is likely to have properties akin to those of the *weight* $w(e) = \mathrm{deg}(v_1) + \mathrm{deg}(v_2)$. It is a result of Borodin [1] that every planar graph under consideration has a $(5,6)$ edge, a $(4,7)$ edge, a $(3,10)$ edge, or an edge with degrees less than one of these. As a direct consequence of this:

**Theorem.** Every planar graph $G$ has an edge $e$ with $D(e) \leq 30$.

**Question 2:** Is there a reference for this result?

I am interested in the values of $D$ that can and must appear in a graph with $n$ vertices.

**Definition.** Let $f(n)$ be the least natural number $N$ such that every planar graph with $n$ vertices has an edge $e$ with $D(e) \leq N$.

The theorem above implies that for all $n$, $f(n) \leq 30$. For example, $f(4) = 9$ because the only possible graph is the tetrahedron with all vertices degree 3, and so each edge has $D(e) = 9$. I think $f(5)=12$, $f(6)=f(7)=f(8)=16$ and $f(12)=25$. I suspect that $f(n)=30$ for all $n \geq 32$.

**Question 3:** Has this quantity $f(n)$ been studied for any or all $n$?

Thanks for any help.

[1] Borodin, O. V., Joint generalization of the theorems of Lebesgue and Kotzig on the combinatorics of planar maps. (Russian) Diskret. Mat. 3 (1991), no. 4, 24–27.

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