Connected graph with two vertices of odd degrees, not containing an Euler path?

My graphs are undirected and connected and fulfill the above condition. Yet these two graphs have no Eulerian path. Why is it so? graph1 graph2

41 1 1 silver badge 11 11 bronze badges asked Aug 26, 2020 at 7:46 491 4 4 silver badges 17 17 bronze badges $\begingroup$ both of them have 4 vertices with odd degree $\endgroup$ Commented Aug 26, 2020 at 7:48

1 Answer 1

$\begingroup$

These graphs do not have Eulerian paths because they have more than two vertices of odd degree. In this case, both have four vertices of odd degree, which is more than 2.

I have gone through and circled and labeled all of the vertices with odd degree so you can check over which vertices you may have missed.

enter image description here enter image description here

answered Aug 26, 2020 at 8:00 711 4 4 silver badges 22 22 bronze badges $\begingroup$ Thank you. I got confused by the square I believe. $\endgroup$ Commented Aug 26, 2020 at 10:30

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