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Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree. (English) Zbl 0704.05041
Let $$G=(V,E)$$ be a simple graph and it is said to be Class 1 if the chromatic index equals the maximum degree $$\Delta(G)$$. For regular graphs this property means that the edge set is the disjoint union of perfect matchings and such graphs are also called 1-factorizable. The main results of the present paper are three theorems which represent sufficient conditions for simple graphs G to be Class 1 and these conditions essentially depend on the minimum degree $$\delta(G)$$ and on the number k(G) of vertices of maximum degree of G. For instance Theorem 1 reads as follows: Let G be a graph of even order 2n; if G satisfies $$\delta(G)\geq n+k(G)-2$$, then G is Class 1. The proofs of these main theorems are very extensive ones, and by in part repeated applications of well known lemmas they are based on removing a set of perfect and so- called near-perfect matchings from the graph until the number of vertices of maximum degree is atmost 2. In a further section by application of these theorems three conjectures concerning criticality of a graph respectively its being Class 2 respectively its being Class 1 are proved for special cases and with respect to satisfying certain inequalities for $$\Delta(G)$$ and $$k(G)$$. They are consequences of a relative general conjecture of Chetwynd and Hilton, if $$\Delta(G)>\frac{1}{3}| V(G)|$$ is required, and so results on edge-coloring of these authors are improved.

##### MSC:
 05C75 Structural characterization of families of graphs 05C35 Extremal problems in graph theory
##### Keywords:
l-factorization; regular graphs; Class 1
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