# On a {K 4,K {2,2,2}}-ultrahomogeneous graph - Mathematics Combinatorics

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Abstract: The existence of a connected 12-regular $\{K 4,K {2,2,2}\}$-ultrahomogeneousgraph $G$ is established, i.e. each isomorphism between two copies of $K 4$ or$K {2,2,2}$ in $G$ extends to an automorphism of $G$, with the 42 orderedlines of the Fano plane taken as vertices. This graph $G$ can be expressed in aunique way both as the edge-disjoint union of 42 induced copies of $K 4$ and asthe edge-disjoint union of 21 induced copies of $K {2,2,2}$, with no morecopies of $K 4$ or $K {2,2,2}$ existing in $G$. Moreover, each edge of $G$ isshared by exactly one copy of $K 4$ and one of $K {2,2,2}$. While the linegraphs of $d$-cubes, $3\le d\in\ZZ$, are $\{K d, K {2,2}\}$-ultrahomogeneous,$G$ is not even line-graphical. In addition, the chordless 6-cycles of $G$ areseen to play an interesting role and some self-dual configurations associatedto $G$ with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs areconsidered.

Autor: Italo J. Dejter

Fuente: https://arxiv.org/