IR-TST 2021 Problem 2

In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$ . Let $d>3$ be the biggest degree in the graph $G$ . Prove that if : $$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$ Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected.

Proposed by Ali Mirzaie