On the properties of weighted minimum colouring games

Herbert Hamers (), Nayat Horozoglu (), Henk Norde () and Trine Platz
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Herbert Hamers: Tilburg University, and TIAS
Nayat Horozoglu: London School of Economics and Political Science
Henk Norde: Tilburg University

Annals of Operations Research, 2022, vol. 318, issue 2, No 12, 963-983

Abstract: Abstract A weighted minimum colouring (WMC) game is induced by an undirected graph and a positive weight vector on its vertices. The value of a coalition in a WMC game is determined by the weighted chromatic number of its induced subgraph. A graph G is said to be globally (respectively, locally) WMC totally balanced, submodular, or PMAS-admissible, if for all positive integer weight vectors (respectively, for at least one positive integer weight vector), the corresponding WMC game is totally balanced, submodular or admits a population monotonic allocation scheme (PMAS). We show that a graph G is globally WMC totally balanced if and only if it is perfect, whereas any graph G is locally WMC totally balanced. Furthermore, G is globally (respectively, locally) WMC submodular if and only if it is complete multipartite (respectively, $$(2K_2,P_4)$$ ( 2 K 2 , P 4 ) -free). Finally, we show that G is globally PMAS-admissible if and only if it is $$(2K_2,P_4)$$ ( 2 K 2 , P 4 ) -free, and we provide a partial characterisation of locally PMAS-admissible graphs.

Keywords: Weighted minimum colouring game; Totally balancedness; Submodularity; Population monotonic allocation schemes; Complete multipartite graph; $$(2K_{2}$$ ( 2 K 2; $$P_{4})$$ P 4 ) -free graph (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10479-021-04374-9

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