If the core of a graph is a forest, then it is Class 1

It is a standard result, due to Fournier, that if the core of a graph (the induced graph by the vertices having their degrees equal to maximum degree of the graph) is a forest (acyclic), then the graph is Class 1 (edge colorable with colors equal to the maximum degree of the graph.

Is there a simple proof of this fact? Specifically, it is said that it follows from the Vizing Adjacency lemma. Any hints? Thanks beforehand.

• Just reduce to a critical graph and apply the lemma to any leaf. Sep 5, 2019 at 5:54
• @IlyaBogdanov could you elaborate a bit? I am not sure of the full statement of Vizing Adjacency Lemma except that if a graph is edge color critical, it should have $3$ major vertices Sep 5, 2019 at 5:56
• I've googled it quickly;) see, e.g., Theorem 4 in faculty.math.illinois.edu/~kostochk/math581/viz4.pdf Sep 5, 2019 at 6:05
• @IlyaBogdanov thanks! I wrote the answer now Sep 5, 2019 at 6:21

Thanks to the comment by @IlyaBogdanov, consider that the graph is not of Class 1, that is, its edge chromatic number is $$\Delta+1$$, where $$\Delta$$ be the maximum degree of the simple graph. Consider the minimal such graph, that is, a critical graph. Now, this should consist of only two major vertices connected by an edge.