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Item type:Publication, Access status: with files , Fast strategies in waiter-client games(EMIS ELibEMS, 2020-09-18); ; ; ; Waiter-Client games are played on some hypergraph (X,𝓕), where 𝓕 denotes the family of winning sets. For some bias b, during each round of such a game Waiter offers to Client b+1 elements of X, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from 𝓕. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. X=E(Kn), in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Publicationtype: Journal ArticleScopus© Citations 2 238 280 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , On minimal Ramsey graphs and Ramsey equivalence in multiple coloursFor an integer q≥ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H, yet no proper subgraph of G has this property then G is called q-Ramsey-minimal for H. Generalising a statement by Burr, Nešetřil and Rödl from 1977 we prove that, for q≥ 3, if G is a graph that is not q-Ramsey for some graph H then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H, as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences. (1) For 2≤ r< q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H. (2) For every q≥ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus, and chromatic number. (3) The collection { 𝓜q(H) : H is 3-connected or K₃} forms an antichain with respect to the subset relation, where 𝓜q(H) denotes the set of all graphs that are q-Ramsey-minimal for H. We also address the question which pairs of graphs satisfy 𝓜q(H₁)= 𝓜q(H₂), in which case H₁ and H₂ are called q-equivalent. We show that two graphs H₁ and H₂ are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q≥ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.Publicationtype: Journal Article152