Up-embeddability of graphs with small order

Let GG be a 2-edge connected simple graph with girth gg and minimal degree δ≥3δ≥3. If one of the following conditions is satisfied: (1) |V(G)|<3M(δ,g)|V(G)|<3M(δ,g); (2) when GG is 3-edge connected, δ=3δ=3 and |V(G)|<6M(3,g)−6|V(G)|<6M(3,g)−6; or, δ≥4δ≥4 and |V(G)|<6M(δ,g)|V(G)|<6M(δ,g), then GG is up-embeddable. Here, M(δ,g)M(δ,g) is the Moore bound of the (δ,g)(δ,g)-cage. As a corollary, there exists a constant cc such that when δ>|V(G)|−6cr+1(r=⌊g−12⌋), GG is up-embeddable.