Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.