# Indirect Proofs

An indirect proof is proof that an implication, pq, is true by showing that the contrapositive of the implication ¬q ⇒ ¬p is true. That is, the indirect proof would assume that ¬q is true and argue that ¬p must also be true. Recall that an implication and its contrapositive are equivalent. So, proving that ¬q ⇒ ¬p also proves that pq.

Example 1: Consider this very simple claim about numbers:
Claim: Let n be an integer. If 5n + 4 is even, then n is even.

Proof: We will prove this claim indirectly. So, suppose that n is odd. Then, there must be an integer k such that n = 2k + 1. So, we have
5n + 4 = 5 (2k + 1) + 4 = 10 k + 9 = 2 (5k + 4) + 1
Thus, 5n + 4 = 2 m + 1 where m = 5k + 4. Therefore, 5n + 4 is an odd integer.
QED

Example 2:
Claim: If a graph G has an Euler circuit, then every vertex in G has even degree.

Proof: Suppose that G has a vertex v with odd degree. Let π be any circuit in G. If π does not visit v, then π is not an Euler circuit since it misses all edges of the edges incident on v. (Since v has odd degree, there is at least one edge incident on v.) Let k be the number of times that v appears on the circuit π. Each time that π visits v, it must enter v using an edge and leave v using a different edge. If any of these edges are repeated, then π is not an Euler circuit. Thus, π uses exactly 2k edges incident on v. Since 2k is an even number and v has odd degree, π must miss at least one edge incident on v. Thus, π is not an Euler circuit and G does not have any Euler circuits.
QED

Using an indirect proof allows us to assume the existence of a vertex v with odd degree. We can then use this "fact" to conclude that none of the circuits in G are Euler circuits.

Note: the proof above assumes that v does not have any self-loops (that is, an edge from v to itself). As discussed in class, self-loops are inconsequential since a graph has an Euler circuit if and only if the graph with all self-loops removed has an Euler circuit.