Discrrete mathematics for computer science coloring warmup

Coloring Warm-Up


A graph is 2-colorable iff it has no odd length
cycles
1: If G has an odd-length cycle then G is not 2-colorable
Proof: Let v0, …, v2n+1 be an odd length cycle (n≥1). Suppose G
can be colored with two colors red and blue. Without loss of
generality [WLOG] color v0 red.
Then v1 must be colored blue since v0 and v1 are adjacent. Then
v2 must be colored red, and in general, all even-indexed vertices
must be colored red and all odd-indexed vertices must be
colored blue.
But v0=v2n+1 must get both colors, contradiction.


A graph is 2-colorable iff it has no odd length
cycles
2: If G has no odd-length cycles then G is 2-colorable.
How to define the coloring?
Helpful Def. The distance of one vertex from another is the length of
the shortest path between them.
Proof: Pick any connected component.
Pick an arbitrary vertex v. Color all vertices at even distance from v
with one color and all vertices at odd distance from v with the
other color.


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It suffices to prove that for every n, there are no nodes in G that
(a) are distance n apart but (b) also have a path between them
of length m, where one of m, n is even and the other is odd.
Suppose there were. By WOP there is a least such n, and such a
pair of nodes v, w.
Then the two paths have no vertices in common except the
beginning and end.
But the round trip out on one path and returning on the other
path is a cycle of length m+n, which is odd, contradiction. QED