Let an $i$-vertex be a vertex of degree $i$. Let an $i, j-$ edge be an edge joining an $i-$vertex to a $j-$vertex. Given a plane graph G, let $e_{i,j}$ be the number of $i, j-$edges of $G$.

I found **Borodin** and **Sanders**'s work through the following paper:

- Oleg V. Borodin, Daniel P. Sanders,
*On Light Edges and Triangles in Planar Graphs of Minimum Degree Five*, Math. Nach.**170**Issue 1 (1994) pp 19–24, doi:10.1002/mana.19941700103.

**Theorem** Every normal plane map of minimum vertex degree $5$ satisfies

$$\frac{14}{3}e_{5,5}+2e_{5,6}\ge 120.$$
The author says the inequality **cannot be improved**.
Since the following graph is constructed with $e_{5,5}=24$ and $e_{5,6}=4$.

I am very confused about the reason of "cannot be improved". More clearly the author's graph makes me unconvinced. Because we are not clear about the edges outside this local subgraph $H$. It is still possible that $e_{5,5}>24$ and $e_{5,6}>4$ . If this happens, it seems that this lower bound can be improved.

My point is only if we can construct a graph with $n$ (**arbitrarily large**)vertices with minimum degree $5$, but
$$\frac{14}{3}e_{5,5}+2e_{5,6}= 120.$$
Only then we can say that the coefficient cannot be improved. Otherwise we might even get
$$ke_{5,5}+le_{5,6}\ge f(n),$$
where $k$ and $l$ are all constant, and $f(n)$ is function about $n$.

After all, there are 5-regular planar graph with $e_{5,5}=\frac{5n}{2}$ by doing limited copy of following graph. (Regrettably, I haven't thought of the connected graph example with $n$ ($ n>12$) vertices.It will be better.)

I don’t know if there is a problem with my thinking, I would like to get some guidance.Thanks in advance!