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 Ḡ. 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. © 2004 Elsevier B.V. All rights reserved.
Stehlík, M. (2004). A generalisation of matching and colouring. Discrete Mathematics, 285(1–3), 257–265. https://doi.org/10.1016/j.disc.2004.04.006