Discrrete mathematics for computer science coloring warmup

Suppose G can be colored with two colors red and blue.. 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

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

• 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