, If {a, b, c} ? X 4?j , we may assume that ?(a) = 1, ?(b) = 2 and ?(c) = 3. There is a neighbour d of c with ?(d) = 2 and a neighbour e of d with ?(e) = 1. Since {a, e} and {b, d} are stable sets in H k , we have d ? X 2 , e ? X 4?j . But then the path e-d-c-u-a contradicts property (2), {v, w} and j the index such that t ? X j . Vertex u has three neighbours a, b, c such that {?(a), ?(b), ?(c)} = {1, 2, 3}. By property (1), vol.4

, V (G) = {v 1 , v 2 }. We claim that ?(G × H k ) ? k. To see this, let Y i = {v 1 claim that ?(G H k ) ? k. To see this, let Y i = {v i } × X i for i ? {1, vol.2, p.3

, If G is a the disjoint union of complete graphs, then there is an upper bound on ?(G H) as a function of ?(G) and ?(H)

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