An adjacency lemma for critical multigraphs

In edge colouring it is often useful to have information about the degree distribution of the neighbours of a given vertex. For example, the well-known Vizing's Adjacency Lemma provides a useful lower bound on the number of vertices of maximum degree adjacent to a given one in a critical graph. We consider an extension of this problem, where we seek information on vertices at distance two from a given vertex in a critical graph. We extend and, simultaneously, generalize to multigraphs two results proved, respectively, by Zhang [Every planar graph with maximum degree seven is Class 1, Graphs Combin. 16 (2000) 467–495] and Sanders and Zhao [Planar graphs of maximum degree seven are Class 1, J. Combin. Theory Ser. B 83 (2001) 201–212].