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• W3081230275 abstract "For a pair of natural numbers k,l, a (k,l)-colouring of a graph G is a partition of the vertex set of G into (possibly empty) sets S1,S2,…,Sk, C1,C2,…,Cl such that each set Si is an independent set and each set Cj induces a clique in G. The (k,l)-colouring problem, which is NP-complete in general, has been studied for special graph classes such as chordal graphs, cographs and line graphs. Let κˆ(G)=(κ0(G),κ1(G),…,κθ(G)−1(G)) and λˆ(G)=(λ0(G),λ1(G),…,λχ(G)−1(G)) where κl(G) (respectively, λk(G)) is the minimum k (respectively, l) such that G has a (k,l)-colouring. We prove that κˆ(G) and λˆ(G) are a pair of conjugate sequences for every graph G and when G is a cograph, the number of vertices in G is equal to the sum of the entries in κˆ(G) or in λˆ(G). Using the decomposition property of cographs we show that every cograph can be represented by Ferrers diagram. We devise algorithms which compute κˆ(G) for cographs G and find an induced subgraph in G that can be used to certify the non-(k,l)-colourability of G." @default.
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• W3081230275 date "2021-01-01" @default.
• W3081230275 modified "2023-09-23" @default.
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• W3081230275 doi "https://doi.org/10.1016/j.ejc.2020.103208" @default.
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