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 $S_0 = \{ r \}$.

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, $V = \displaystyle \bigcup_{n \in \N} S_n$.

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

Define a relation $\mathcal R$ on $V$ by letting $p \mathrel{\mathcal R} q$ Definition:iff:


 * $p$ 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 $\mathcal R^+(v)$ is infinite.

$r \in V'$:

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 $v \in \mathcal R^+(r)$.

As this holds for all $v \in V$, and $V$ is infinite, $\mathcal R^+(r)$ is infinite, so $r \in V'$.

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

$\mathcal R'$ is a left-total relation: If $v \in V'$, then $\mathcal R^+(v)$ is infinite. Since $\mathcal R(v)$ is finite, Finite Union of Finite Sets is Finite shows that $\mathcal R(v)$ must have an element $u$ such that $\mathcal R^+(u)$ 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:


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

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