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关于{K_4，K_{2,2,2} }-超齐次图

On a {K_4,K_{2,2,2}}-ultrahomogeneous graph

Italo J. Dejter

备注：
12 pages, 4 figures
发表信息：
Australasian Jour. of Combinatorics, 44 (2009), 63--75
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摘要：

以Fano平面上的42条有序线为顶点，证明了连通12-正则$\{K_4,K_{2,2,2}\}$ -超齐次图$G$的存在性（即$G$中两个$K_4$或$K_{2,2,2}$之间的同构可扩展为$G$的一个自同构）.该图$G$可以以独特的方式表示为42个诱导的$K_4$拷贝的边不相交的并集和21个诱导的$K_{2,2,2}$拷贝的边不相交的并集，在$G$中不存在更多的$K_4$或$K_{2,2,2}$拷贝。此外，$G$的每条边恰好由$K_4$的一个拷贝和$K_{2,2,2}$的一个拷贝共享。虽然$d$ -立方体的线图（$3\le d\in\ZZ$）是$\{K_d, K_{2,2}\}$ -超均匀的，但$G$甚至不是线图。此外，我们还研究了$G$的无弦6-圈，并讨论了与$G$相关的具有2-弧传递、弧传递和半对称Levi图的自对偶结构.

英文摘要：

The existence of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph $G$ 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 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 $K_4$ and as the edge-disjoint union of 21 induced copies of $K_{2,2,2}$, with no more copies of $K_4$ or $K_{2,2,2}$ existing in $G$. Moreover, each edge of $G$ is shared by exactly one copy of $K_4$ and one of $K_{2,2,2}$. While the line graphs 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 $G$ 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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