Clebsch-Graph

Later some confusion has arisen, and some authors use the name "Clebsch graph" for the complement of Γ. The Clebsch graph is the halved 5-cube, that is, the vertices are the binary vectors of length 5 and even weight, joined when the Hamming distance is 2. The Clebsch graph is the graph obtained from K1+T(6) by switching w.r.t. the set of 10 pairs not containing a fixed symbol (see also 2-graphs ). Equivalently, the complement of the Clebsch graph is the graph obtained from the 4-cube by joining antipodes by an edge. The complement of the Clebsch graph is the graph on GF(16) where two points are adjacent when their difference is a cube. It follows that K16 is the edge-disjoint union of three copies of the complement of the Clebsch graph. The Clebsch graph is the local graph of the Schläfli graph . The Clebsch graph has independence number 2 and chromatic number 8. The complement of the Clebsch graph has independence number 5 and chromatic number 4. A. Clebsch, Ueber die Flächen vierter Ordnung, welche eine Doppelcurve zweiten Grades besitzen , J. für Math. 69 (1868) 142-184. W.H. Clatworthy, Partially balanced incomplete block designs with two associate classes and two treatments per block , J. Res. Nat. Bur. Standards 54 (1955) 177-190.