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Balakrishnan, R. and Francis Raj, S. , Bounds for the b-chromatic number of the Mycielskian of some families of graphs, to appear in Ars. Combinatoria . Balakrishnan, R

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-chromatic number of a graph, Discrete Applied Mathematics 91 (1999), 127–141. MR 2000a :05079 Manlove D. F. The b-chromatic number of a graph

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Abstract  

We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Karłowicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some aXY the order-type of aX is different from that of aY. In addition to the chromatic number x(G) of these graphs we study χκ(G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.

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Abstract  

We determine a class of triple systems such that each must occur in a triple system with uncountable chromatic number that omits
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(the unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also show that consistently there are two finite triple systems such that they can separately be omitted by uncountably chromatic triple systems but not both.
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References [1] Balakrishnan , R. and Francis Raj , S. , Bounds for the b-chromatic number of G - v , Discrete Appl. Math

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Acta Mathematica Hungarica
Authors: András Hajnal, István Juhász, Lajos Soukup, and Zoltán Szentmiklóssy

] Erdős , P. , Galvin , F. , Hajnal , A. 1975 On set-systems having large chromatic number and not containing prescribed subsystems Infinite and Finite Sets Colloq. Keszthely 1973 Colloq. Math. Soc. János Bolyai 10 North-Holland Amsterdam 425 – 513

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