On rainbow tetrahedra in Cayley graphs

Let $\Gamma_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $\Gamma_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations. Generalizing asymptotic results are discussed as well.Comment: 21 pages, 7 figure

On Two Ways of Enumerating Ordered Trees

The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient pseudograph $R_k$ whose vertices are the $k$-edge ordered trees $T$, each $T$ encoded as a $(2k+1)$-string $F(T)$ formed via $\rightarrow$DFS by: {\bf(i)} ($\leftarrow$BFS-assigned) Kierstead-Trotter lexical colors $0,\ldots,k$ for the descending nodes; {\bf(ii)} asterisks $*$ for the $k$ ascending edges. Two ways of corresponding a restricted-growth $k$-string $\alpha$ to each $T$ exist, namely one Stanley's way and a novel way that assigns $F(T)$ to $\alpha$ via nested substring-swaps. These swaps permit to sort $V(R_k)$ as an ordered tree that allows a lexical visualization of $M_k$ as well as the Hamilton cycles of $M_k$ constructed by P. Gregor, T. M\"utze and J. Nummenpalo.Comment: 26 pages, 8 figures, 10 table

On a C⃗4\vec{C}_4-ultrahomogeneous oriented graph

The notion of a $\mathcal C$-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and subsequently a strongly connected $\vec{C}_4$-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in which pencils are replaced by ordered pencils.Comment: 4 pages, 2 figures, 2 table