2007/04/30 by Italo J. Dejter · 1 citation
Mathematics · #math.CO #msc:05C62
published as Australasian Jour. of Combinatorics, 44 (2009), 63--75 · 12 pages, 4 figures
arxiv created 2008/10/20 · arxiv updated 2009/12/01
The existence of a connected 12-regular \K4,K2,2,2\-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 induced copies of K2,2,2, with no more copies of K4 or K2,2,2 existing in G. Moreover, each edge of G is shared by exactly one copy of K4 and one of K2,2,2. While the line graphs of d-cubes, (3≤ d∈\ZZ), are \Kd, K2,2\-ultrahomogeneous, G is not even line-graphical. In addition, the chordless 6-cycles of G are seen to play an interesting role and some self-dual configurations associated to G with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered.