In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph $G$, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of $G$ is 2-arc-connected. Finally, we characterize all 2-connected locally finite outerplanar graphs nonisomorphic to $K_3$ as precisely those graphs such that every vertex is incident to exactly two contractible edges as well as those graphs such that every finite bond contains exactly two contractible edges.

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One of the most famous open problems in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In this paper, we define graceful labelings for countably infinite graphs, and state and verify a Graceful Tree Conjecture for countably infinite trees.

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