Up-embeddability of graphs with small order

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