Let $G$ be a connected graph such that $δ(G)≥k$. Proof there exists a path $P$ of length $k$ such that $G-P$ is connected.
I am not sure how do I actually approach this question. Do I consider contradiction or inductive proof, if so how. Really appreciate it alot
We will try to prove that the path (v1,v2,...,vk) exist for every vertex v1 in G.
Let v1 be a vertex in G, then deg(v1)≥k since δ(G)≥k.
next we will pick a vertex from the k possible vertices, let call it v2 again deg(v2)≥k but we only have k-1 possible vertices we choose again a vertex v3 again deg(v3)≥k but we only have k-2 possible vertices.
since the graph is connected the path exists since their is a path between every pair of vertices.