Some of your friends work on wireless networks and they're currently studying the properties of a network of n mobile devices. As the devices move around (actually as their human owners move around), they define a graph at any point in time as follows: there is a node representing each of the n devices, and there is an edge between device i and device j is the physical locations of i and j are more than 500 meters apart. (If so, we say that i and j are "in range" of each other.)

They'd like it to be the case that the network of devices connected at all times, and so they've constrained the motion of the devices to satisfy the following property: each device i is within 500 meters of at least n/2 of the other devices. (We'll assume n is an even number.) What they'd like to know is: Does this property by itself guarantee that the network will remain connected?

here's a concrete way to formulate the question as a claim about graphs.

Claim: Let G be a graph on n nodes, where n is an even number. If every node of G has degree at least n/2, then G is connected.

Decide whether you think the claim is true or false and give proof of either the claim or negation