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This paper is the …rst article in a series devoted to the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this 'precoloring' be extended to a proper coloring of G with at most k colors (for some given k)? This question was motivated by practical problems in… (More)

We continue the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this " precoloring " be extended to a proper coloring of G with at most k colors (for some given k)? Here we investigate the complexity status of precoloring extendibility on some classes of perfect… (More)

A complete k-coloring of a graph G = (V, E) is an assignment ϕ : V → {1,. .. , k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in… (More)

We survey some combinatorial results which are all related to some former results of ours, and, at the same time, they are all related to the famous K˝ onig–Egerváry theorem from 1931.

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