A 3D analog of problem M for a third-order hyperbolic by Volkodavov V.F., Radionova I.N., Bushkov S.V.

By Volkodavov V.F., Radionova I.N., Bushkov S.V.

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5 Generalized perfection There are many generalizations of the parameters \,u,a, and K of a graph; the generalized parameters also satisfy the min-max inequalities. This leads to several concepts containing perfection as a special case. The first such generalization is due to Hell and Roberts [538], and is based on the following notion due to Harary [519] and Sabidussi [939]. 1 Let G and G' be graphs. The lexicographic product GoG' is the graph with vertex set V(G) x V(G') and edge set {(9i,9()(92,92) • 9192 € E(G) V (9l = g2 A && e E(G'))}.

6. 6 Let G be a graph and i be a positive integer. A Ki+\ -free k-coloring of G is a partition of the vertex set of G into k subsets each of which induces a Ki+\-free subgraph of G. Xi(G) denotes the smallest number k for which G has a Ki+\-free k-coloring. (G — G')=u(G)-i. 3 For all positive integers i, all perfect graphs G satisfy Xi(G) = w;(G) and have a min{u>(G),i}-transversal. See also the remarks at the end of the next section.

Note that for every graph G and set S C Ci(G), Ci(S) > Pi(3). This leads to a quantification over all subsets of the set Ki(G). 7 Let i > 2 be an integer. A graph G is AVperfect if for each

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