A generalisation of matching and colouring

Abstract

We define an r- bounded cover of a graph G to be a subgraph T⊆ G such that T contains all vertices of G and each component of T is a complete subgraph of G of order at most r. A 2-bounded cover of a graph G corresponds to a matching of G, and an ω( G)-bounded cover of G corresponds to a colouring of the vertices of the complement G ̄ . We generalise a number of results on matching and colouring of graphs to r-bounded covers, including the Gallai–Edmonds Structure Theorem, Tutte's 1-Factor Theorem, and Gallai's theorem on the minimal order of colour-critical graphs with connected complements.