Euler Formula and its Version
Duration: 4 min
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The video is a lecture on deriving inequalities for connected planar graphs using Euler's formula. The instructor, Sanchit Jain, explains how to derive two specific inequalities: $e \le 3v - 6$ and $r \le 2v - 4$. He starts by listing four points on a slide, explaining the relationship between edges and faces ($2e \ge 3r$) and the definition of region degree. He then proceeds to algebraically derive the first inequality by substituting Euler's formula into the edge-face inequality.
Chapters
0:00 – 2:00 00:00-02:00
The instructor begins by presenting a slide titled 'Other formula derived from Euler's formula'. He lists four key points. Point 1 states that connected planar graphs with more than one edge obey $2e \ge 3r$. He explains the logic: each face has at least three face-edge incidences, and each edge contributes exactly two incidences. Point 2 defines the degree of a region as the number of edges covering it, noting the sum of degrees equals $2|E|$. He then focuses on Point 3, which claims that eliminating $r$ from $r = e - v + 2$ and $3r \le 2e$ results in $e \le 3v - 6$. He starts the derivation on the board, writing $r = e - v + 2$ as equation (1) and $r \le rac{2}{3}e$ as equation (2). He substitutes (1) into (2) to get $ rac{2}{3}e \ge e - v + 2$. Multiplying by 3 gives $2e \ge 3e - 3v + 6$. Rearranging terms leads to $-e \ge -3v + 6$, and finally $e \le 3v - 6$, which he boxes.
2:00 – 4:06 02:00-04:06
Next, the instructor derives the formula for Point 4. He writes Euler's formula $r = e - v + 2$ again. He rearranges it to solve for $e$, obtaining $e = r + v - 2$. He then uses the inequality $3r \le 2e$, which can be rewritten as $e \ge rac{3}{2}r$. Substituting the expression for $e$ into this inequality gives $r + v - 2 \ge rac{3}{2}r$. To clear the fraction, he multiplies the entire inequality by 2, resulting in $2r + 2v - 4 \ge 3r$. He subtracts $2r$ from both sides to get $2v - 4 \ge r$, or $r \le 2v - 4$. He boxes this final result and points to the corresponding line on the slide, completing the derivation of the second inequality.
The lecture provides a rigorous algebraic derivation of two critical inequalities for connected planar graphs. Starting with the basic property that $2e \ge 3r$ due to face-edge incidences, the instructor combines this with Euler's formula ($r = e - v + 2$). By substituting and rearranging terms, he first eliminates the variable $r$ to prove that the number of edges is bounded by $e \le 3v - 6$. Subsequently, he eliminates $e$ to prove that the number of regions is bounded by $r \le 2v - 4$. These steps are clearly written on the whiteboard, showing the transition from Euler's formula to these specific constraints, which are essential for determining if a graph is planar.