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 L. Zhang [Every planar graph with maximum degree seven is Class 1, Graphs and Combinatorics (2000) 16: 467-495 ] and D.P. Sanders and Y. Zhao [Planar graphs of maximum degree seven are Class 1, JCTB (2001) 83: 201-212 ].