On the chromatic number of cube-like graphs. (English) Zbl 0772.05043

A cube-like graph is a graph whose vertices are all \(2^ n\) subsets of a set \(E\) of cardinality \(n\) and in which two vertices are adjacent if their symmetric difference is a member of a given specified collection of subsets of \(E\). This paper shows a cube-like graph of chromatic number 7 and provides that no cube-like graph has chromatic number 3.


05C15 Coloring of graphs and hypergraphs
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[1] Dvorak, T.; Havel, I.; Laborde, J. M.; Liebl, P., Generalized hypercubes and graph embedding with dilation, Math. Kolloq., 40 (1989)
[2] Harary, F., Four difficult unsolved problems in graph theory, (Recent Advances in Graph Theory (1974), Academia: Academia Praha), 249-256 · Zbl 0329.05125
[3] Hebbare, R.; Laborde, J. M., Another characterization of hypercubes, Discrete Math., 39, 161-166 (1982) · Zbl 0482.05033
[4] F. Jaeger, private communication.; F. Jaeger, private communication.
[5] Linial, N.; Meshulam, R.; Tarsi, M., Matroidal bijections between graphs, J. Combin. Theory Ser. B, 45, 31-44 (1988) · Zbl 0724.05067
[6] Mulder, H. M., (0,λ)-graphs and \(n\)-cubes, Discrete Math., 28, 179-188 (1979) · Zbl 0418.05034
[7] Sokolova, M., The chromatic number of extended odd graphs is four, Časopis Pěst. Mat., 112, 308-311 (1987) · Zbl 0667.05023
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