Spanning even trees of graphs

AbstractA tree is said to be even if all pairs of vertices of degree one in are joined by a path of even length. We conjecture that every ‐regular nonbipartite connected graph has a spanning even tree and verify this conjecture when has a 2‐factor. Well‐known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when is odd and has at least bridges. We investigate this case further and propose some related conjectures.