Variations on the Roy-Gallai theorem

Abstract.

A generalization of the Roy-Gallai Theorem on the chromatic number of a graph is derived which is also an extension of several other results of Berge and of Li. A simple inductive proof is given which provides a direct way of deriving the Theorem of Li. We also show that some classical results valid for optimal colorings cannot be transposed to suboptimal colorings. We finally investigate some elementary properties which are also valid in suboptimal colorings.