Bipartite Graphs and Matchings · Medium problem

The Petersen graph has an outer $5$-cycle on vertices $1,2,3,4,5$ (edges $12,23,34,45,51$), an inner $5$-cycle on vertices $6,7,8,9,10$ connecting each to the vertex two steps ahead ($68, 79, 8\text{-}10, 9\text{-}6, 10\text{-}7$), and $5$ spokes $16, 27, 38, 49, 5\text{-}10$. What is the length of the shortest cycle in this graph that has odd length?