Showing papers by "Norbert Sauer published in 1990"

Partition Theorems for Graphs Respecting the Chromatic Number

Abstract: We exhibit general results which allow us to deduce the following: For every family G 1, G 2,…, G m of K r -free graphs there exists a K r -free graph G with χ(G) = χ(G 1) + χ(G 2)+…+χ(G m ) - m + 1 such that for every partition of the vertices of G, V(G) = V 1 \(V\left( G \right) = {V_1}\dot U{V_2}\dot U \ldots \dot U{V_m}\) V 2 \(V\left( H \right) = {V_1}\dot U \ldots \dot U{V_{n + 1}}\)…\(V\left( H \right) = {V_1}\dot U \ldots \dot U{V_{n + 1}}\) V m there exists an Ii and an embedding α : G i → H such that α(G i ) ⊆ V i . And for each n-chromatic K r -free graph G, there exists an n-chromatic K r -free graph H such that for every partition of the vertices of H, V(H) = V1 \(V\left( H \right) = {V_1}\dot U \ldots \dot U{V_{n + 1}}\)…\(V\left( H \right) = {V_1}\dot U \ldots \dot U{V_{n + 1}}\) V n+1, there exists an i with 1 ≤ i ≤ n + and an embedding σ: G → H with σ(G) ∩ V i = O.