An Euler trail is a walk that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. A Euler trail with same endpoints is a Euler circuit. Here are some examples: Letbe a finite connected graph and denote the number of vertices that have odd degrees. Prove that G has a Euler trail if and only if . First, by the Handshake Theorem, we c..
Prove that for , . Intuition: Suppose n people shake hands with other people in the group. They don't shake hands twice with the same person.How many handshakes take place?각 사람을 노드로 본다면 두 사람이 악수를 하면 그 두 노드 사이에 Edge 가 생긴다.전체 악수의 수 = Number of Edges in the graph = 각 사람이 한 악수의 수의 총합 / 2 = Sum of degree of every vertex / 2 Let be the vertices of G and their degrees respectively. Also, Then we know t..