I have a conjecture about odd cycle in a simple graph,but I can not proof it or find a counterexample.So I want to ask for some help.My conjecture is:
Let k be a positive integer and G be a 2-connected simple gragh satisfy:
(1)G is not complete;
(2)For every two different vertices u and v of G which are not adjacent, there exists a path link u and v whose length is odd and not less then 2k+1.
Then there must exists an odd cycle in G whose length is not less than 2k+1.