1

enter image description here

Petersen's Graph

Is there a path of length $10$ in the Petersen's graph? If so, is it true that Petersen's graph would be Hamiltonian? Is the fact that each vertex has degree $3$ fundamental to disprove the previous statement; because if I start at a vertex $v_0$, then I should come back to it by the end of my path, but this would imply that we didn't pass by one edge that is one of three incident edges to the vertex $v_0$. Any suggestions?

J P
  • 355
  • 2
    By a cyclic graph do you mean a Hamiltonian graph? Petersen's graph is not a cycle graph, and this is easy to see just by the fact that its vertices do not have degree $2$. – Misha Lavrov Nov 14 '23 at 19:54
  • Note: D. West proved that the Petersen graph is nonhamiltonian. – Anton Vrdoljak Nov 14 '23 at 19:58
  • @AntonVrdoljak With all respect to D. West, that can't possibly be right; he was born in 1953, which is 55 years after Julius Petersen proved that it has no $3$-edge-coloring, and that is a strictly stronger statement. (There was less interest in Hamiltonian cycles at the time, but I do not believe the observation stood unmade for that long.) Now, if you simply mean that West's Introduction to Graph Theory presents a proof that the Petersen graph is not Hamiltonian, that is true. – Misha Lavrov Nov 14 '23 at 21:20
  • 1
    What is the degree of a path?? Is "path of degree 10" a typo for "cycle of length 10"? – bof Nov 14 '23 at 21:26
  • To @MishaLavrov: True, my note was regarding to that reference. Thanks for your comment. – Anton Vrdoljak Nov 15 '23 at 06:17

0 Answers0