Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without u...
8. This question has two parts. (i) Let G be a graph with minimum vertex degree 8(G) > k. Then prove that G has a path of length at least k. (ii) Let G be a graph of order n. If S(G) > nz?, then prove that G is connected.
Use induction on n... 5. Use induction on n to prove that any tree on n2 2 vertices has at least two vertices of degree 1 (a vertex of degree 1 is called a leaf). 5. Use induction on n to prove that any tree on n2 2 vertices has at least two vertices of degree 1 (a vertex of degree 1 is called a leaf).
please solve without using Konig theorem Let G be a bipartite graph of order n. Prove that a(G) = if and only if G has a perfect matching.
1. Use Pigeon hole principle to prove that any graph with at least 2 vertices contains two vertices of the same degree. (Hint: Prove by contradiction. (4 points) 2. Given (6 Points) a. Prove the above equation using binomial theorem. (3 Points) b. Give a combinatorial proof for the given equating. (3 Points) 4n = (0)2" + (1)2" +...+)2"-
Problem 2.13 - page 31. Let G be an n-vertex graph such that for any non-adjacent vertices U, V EV(G), d(u) + d(u) > n. Prove that G is Hamiltonian
Question 4 (20 points) Let F: R R be any homogeneous polynomial function (with degree no less than one) with at least one positive value. Prove that the function f:Rn R, f(x) F(x) 1, defines on f-1(0) a structure of smooth manifold.
need help with a and b in this graph theory question Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and connecting each vertex to the exact a) Show that for all u,v there are k internally disjoint u, v-paths (you (b) Use the previous part, even if you did not prove it, to show that the e vertex and the k 1 closest vertices on either...
Problem 3. Read about compactness in Section 2.8 of the book. Then, prove, WITHOUT RELYING ON HEINE-BOREL's THEOREM, the following. Let E be a closed bounded subset of E and r be any function mapping E to (0,00). Then there ensts finitely many pints yi E E,i = 1, , N such that i-1 Here Br(y.)(y) is the open ball (neighborhood) of Tudius r(y.) centered at yi. Problem 3. Read about compactness in Section 2.8 of the book. Then, prove,...
Please prove the theorems, thank you 6.1 Theorem. Let anx+an-1- +ag he a polynomial of degree n0 with integer coefficients and assume an0. Then an integer r is a Poot of (x) if and only if there exists a polynomlal g(x) of degree n - with integer coeficients such that f(x) (x)g(x). This next theorem is very similar to the one above, but in this case (xr)g(x) is not quite equal to f(x), but is the same except for the...
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...