in Graph Theory
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why in this planar graph this theorem ,”sum of degrees of faces or regions is twice the number of edges” is not true as it should hold for all planar graphs??

Note: numbers denote region or face 

 

in Graph Theory
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When an edge lies inside a region, it contributes "2" to the degree of the region it lies within.

Degree of a region is the no.of edges on the boundary.

Degree(region 2)=ab+bc+cd+ad+de+de

Edge de lies in region 2 and contributes 2 to the degree.

So, deg of R2=6.
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Degree of region 2 is $6$ not 4. Maybe you are not counting edge $de$. Also the edge $de$ should be counted twice, bcoz its both sides are exposed to region $2$
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