König's Lemma/Countable

Theorem
Let $G = \left({V, E}\right)$ be a graph with countably infinitely many vertices which is connected and is locally finite.

Then every vertex of $G$ lies on a path of infinite length.

Proof
Let $r$ be a vertex of $G$.

Recursively define a sequence $\left\langle{S_n}\right\rangle$:

Let $S_0 = \left\{ {r}\right\}$.

Let $S_{n+1}$ be the set of all vertices that are adjacent to some element of $S_n$ but not adjacent to any element of $S_k$ for $k < n$.

That is, $S_n$ is the set of vertices whose shortest path(s) to $r$ have $n$ edges.

Since $G$ is connected:
 * $V = \displaystyle \bigcup_{n \mathop \in \N} S_n$

Note that $\left\{ {S_n: n \in \N}\right\}$ is pairwise disjoint.

Define a relation $\mathcal R$ on $V$ by letting $p \mathrel {\mathcal R} q$ :
 * $p$ is adjacent to $q$

and:
 * there exists an $n \in \N$ such that $p \in S_n$ and $q \in S_{n+1}$.

Let $\mathcal R^+$ be the transitive closure of $\mathcal R$.

Let $V'$ be the set of all $v \in V$ such that $\mathcal R^+ \left({v}\right)$ is infinite.

It will be demonstrated that $r \in V'$:

Let $v \in V$.

Since $G$ is connected, there is a path from $r$ to $v$.

By the Well-Ordering Principle, there is a path $P$ from $r$ to $v$ of minimal length.

Then the vertices along $P$ will lie in successive sets $S_k$.

Thus $v \in \mathcal R^+ \left({r}\right)$.

As this holds for all $v \in V$, and $V$ is infinite, $\mathcal R^+ \left({r}\right)$ is infinite.

Therefore $r \in V'$.

Let $\mathcal R'$ be the restriction of $\mathcal R$ to $V'$.

It is to be shown that $\mathcal R'$ is a left-total relation.

Let $v \in V'$.

Then $\mathcal R^+ \left({v}\right)$ is infinite.

Since $\mathcal R \left({v}\right)$ is finite, Finite Union of Finite Sets is Finite shows that $\mathcal R \left({v}\right)$ must have an element $u$ such that $\mathcal R^+ \left({u}\right)$ is infinite.

So $u \in V'$ and $v \mathrel {\mathcal R'} u$.

Since $V$ is assumed to be countably infinite, we can assume WLOG that $V = \N$.

Recursively define a sequence $\langle v_k \rangle$ in $V'$ thus:


 * $v_0 = r$
 * $v_{n + 1} = \min \left({\mathcal R' \left({v_n}\right)}\right)$

Then $\left\langle{v_k}\right\rangle$ is the sequence of vertices along an infinite path starting at $r$.