TITLE: Invariant problem: Prove that you can't color every line of the graph
QUESTION [6 upvotes]: Given a graph with $6$ vertices and $15$ edges, every starting line is colored yellow. On each step we can choose any $3$ lines that create a triangle (triangle's vertices have to match graph's vertices) and change the color of each line. If the color's yellow, change it to red and if it's red change it to yellow. Prove that no matter what you do, you can't get $15$ red lines.


Hint: In order to find invariant choose a vertex and look at the adjacent lines.

Here's the picture:

Well after thinking for a while, I came up with this reasoning: Each vertex has $5$ lines, if the vertex's part of the triangle, then $2$ of it's sides get colored every time. So we want to get $5$ red lines, but we're swapping even lines each time, so we can't reach odd red lines. Therefore we can't color this graph into only red lines.
But I think this requires stronger reasoning. How do I prove this more formally?

REPLY [4 votes]: Yes your proof is perfectly fine, the operation always leaves each vertex with an odd number of yellow edges, hence this number cannot be $0$.