Math 5707 Homework 1
نویسنده
چکیده
Proof: We proved in class (Proposition 2.4.1) that there is at least one triangle-or-anti-triangle, call its vertices {a1, a2, a3}. There are at least three additional distinct vertices, b1, b2, and b3. If all of b1, b2, and b3 are adjacent to each other, then {b1, b2, b3} is a triangle. If none of b1, b2, and b3 is adjacent to any other, then {b1, b2, b3} is an anti-triangle. Assume without loss of generality that neither of these is true. Then there is at least one pair of vertices among b1, b2, and b3 that are not adjacent, and at least one pair that are adjacent. Call the non-adjacent pair u and v and the adjacent pair x and y. Note that {u, v} and {x, y} need not be disjoint. Now, there are two cases to consider: {a1, a2, a3} is either an triangle or an anti-triangle.
منابع مشابه
Math 5707 : Graph Theory , Spring 2017 Homework 5
1.1 Problem Fix a loopless multidigraphD = (V,A, φ). Let f : V → N be a configuration. Let h = ∑ f . Let n = |V |. Assume that n > 0. Let ` = (`1, `2, . . . , `k) be a legal sequence for f having length k ≥ ( n+ h− 1 n− 1 ) . Prove the following: (a) There exist legal sequences (for f) of arbitrary length. (b) Let q be a vertex of D such that for each vertex u ∈ V , there exists a path from u t...
متن کاملMathematics 5707 Homework 1
number of ordered triples (a, b, c) such that ab ∈ E(G) and bc / ∈ E(G). Observe that each vertex v has deg v = 0, 1, 2, 3, 4, or 5. If deg v is 0 or 5, then v is the second vertex in 0 such triples. If deg v is 1 or 4, then v is the second vertex in 4 such triples. If deg v is 2 or 3, then v is the second vertex in 6 such triples. Thus, there are at most 36 such ordered triples. Note that each...
متن کامل