An undirected graph possesses an eulerian circuit if and only if it is…

An Eulerian circuit is a trail in a graph which visits every edge exactly once and starts and ends on the same vertex. For an undirected graph to possess such a circuit, two conditions must be met: the graph must be connected (ignoring isolated vertices), and every vertex in the graph must have an even degree. This is a fundamental result known as Euler's Theorem. When traversing the graph, every time we enter a vertex via an edge, we must leave it via another distinct edge. Therefore, edges come in pairs at each vertex, necessitating an even degree.\nOption (a) correctly states that all vertices must be of even degree, satisfying the theorem. Option (b) is incorrect because odd degrees would imply a vertex has an unmatched edge, preventing a return trip. Option (c) is false since mixed degrees would break the circuit property. Option (d) is irrelevant as the total number of vertices does not determine the existence of an Eulerian circuit. Thus, only condition (a) is required alongside connectivity.