Suppose f:[0,1]->R is continuous, f(0)>0, f(1)=0.
Prove that there is a X0 in (0,1] such that f(Xo)=0 & f(X) >0 for 0<=X<Xo (there is a smallest point in the interval [0,1] which f attains 0)
Since f is continuous, then there exist a sequence Xn converges to X0, and f(Xn) converges to f(Xo)...
Tell me if I'm not on the right track.
Use induction on k.
Pick a path P of maximum length, and suppose vertex vi is a vertex on this path, which has degree at least k, with a set of adjacent vertices {w1,w2,…,wj}, the adjacent vertex set must be on the path.
The minimum path length of a...
This is a graph theory related question.
Let G be a simple graph with min. degree k, where k>=2. Prove that G contains a cycle of length at least k+1.
Am I suppose to use induction to prove G has a path length at least k first, then try to prove that G has a cycle of length at least k+1...
Problem: "Let G be a simple graph on n vertices such that deg(v)>= n/2 for every vertex v in G. Prove that deleting any vertex of G results in a connected graph."
Well, I tried to find the min. case.
Let k be the min. deg. of vertex in a simple graph,
n is number of vertices in G
so k =...
Okay, so by looking at my original assumption P(n-1)=[(n-1)^2/4]=[(n^2-2n+1)/4]=[n^2/4+(1-2n)/4]
So now I need to prove that by adding additional one vertex in result of adding additional (2n-1)/4 edges for all n>5.
So P(n)=P(n-1)+(2n-1)/4= [n^2/4+(1-2n)/4+(2n-1)/4]=[n^2/4]
But how to...
How to prove that the number of edges in a simple bipartite graph with n vertices is at most n^2/4?
Definition of bipartite graph: a graph whose vertex-set can be partitioned into two subsets such that every edge has one endpoint in one part and one endpoint in the other part.
I try to...