The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient graph $R_k$ whose vert... more The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient graph $R_k$ whose vertices represent all $k$-edge ordered (rooted) trees $T$. Each $T$ is encoded as a $(2k+1)$-string $F(T)$ that annotates by means of DFS the descending nodes via Kierstead-Trotter lexical colors $0,\ldots,k$, and the ascending edges via $k$ asterisks. This $T$ is assigned a restricted-growth $k$-string in two ways. One of these two ways is obtained via nested substring-swaps in the $F(T)$'s (i.e. pruning-and-regrafting in the $T$'s). These swaps allow sorting the vertices of all $R_k$'s in a canonical, natural, enumerative way that unifies the presentation of the $M_k$'s and their Hamilton cycles by T. Mütze et al.
A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graphs is gi... more A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graphs is given via their dihedral quotients whose vertices are ordered rooted trees.
Let $0<k\in\mathbb{Z}$. A system of numeration according to which the nonnegative integers are pr... more Let $0<k\in\mathbb{Z}$. A system of numeration according to which the nonnegative integers are presented as restricted growth strings has the $k$-th Catalan number $C_k={2k+1\choose k}/(2k+1)$ presented as the restricted growth string $10^k$. This system yields a linear ordering of the vertex set of a quotient of the graph $M_k$ induced by the $k$- and $(k+1)$-levels of the $n$-cube graph under a natural dihedral group action. Mütze proved the existence of Hamilton cycles of $M_k$. We ask for the existence of Hamilton cycles of $M_k$ with dihedral symmetry so the cited system applies to them.
A system of numeration in which every $k$, with $0<k\in\mathbb{Z}$, appears as a restricted growt... more A system of numeration in which every $k$, with $0<k\in\mathbb{Z}$, appears as a restricted growth string, or RGS, has the $k$-th Catalan number as the RGS $10^k$. This induces a canonical ordering of the vertices of the dihedral quotients of the middle-levels graphs.
Let $\mathcal C$ be a class of graphs. A definition of $\mathcal C$-homogeneous graph $G$ is give... more Let $\mathcal C$ be a class of graphs. A definition of $\mathcal C$-homogeneous graph $G$ is given that fulfills the absence of a fitting generalization of $\mathcal C$-ultrahomogeneous (UH) graph, by considering each induced subgraph of $G$ in $\mathcal C$ as anchored through an arc. Let $2<r\in\Z$, $\sigma\in(0,r-1)\cap\Z$, =2^{\sigma+1}-1$ and $s=2^{r-\sigma-1}$. A construction of non-line-graphical $\{K_{2s},T_{ts,t}}_{K_2}$-homogeneous graphs (meaning ${\mathcal C}=\{K_{2s},T_{ts,t}\}$ with each edge of $G$ shared by exactly one copy of the complete subgraph $K_{2s}$ and one copy of the Turán graph $T_{ts,t}$) is conjectured to yield an infinity of such graphs in terms of configurations of points and lines, one per pair $(r,\sigma)$. Such a construction, based on ordered pencils of binary projective spaces, is shown to yield 21 such graphs not $\{K_{2s},T_{ts,t}\}_{K_2}$-UH graphs. For $s=2$, these are shown to be $K_4$-UH graphs, amounting to six of the 21 graphs, for $r=\sigma+2$ and $\sigma=1,...,6$, with orders $(2^r-1)(2^r-2)$, namely 42, 210, 930, 3906, 16002, 64770, and numbers $4(2^{\sigma-1})$ of edge-disjoint copies of $K_4$ incident to each vertex, namely 4, 12, 28, 60, 124, 252, respectively.
A connected edge-disjoint union $\mathcal Y$ of 102 tetrahedra (copies of $K_4$) is constructed w... more A connected edge-disjoint union $\mathcal Y$ of 102 tetrahedra (copies of $K_4$) is constructed which is Menger graph of a self-dual $(102_4)$-configuration and $K_4$-ultrahomogeneous. As $\mathcal Y$ is not a line graph, we ask whether there exists a non-line-graphical $K_4$-ultrahomogeneous Menger graph of a self-dual $(n_4)$-configuration which is connected edge-disjoint union of $n$ copies of $K_4$, for $n\notin\{42,102\}$. Moreover, $\mathcal Y$ is union of 102 cuboctahedra (copies of $L(Q_3)$) with no two sharing a chordless 4-cycle and has an $L(Q_3)$-ultrahomogeneity property restricted to preserving an edge partition of each $L(Q_3)$ into 2-paths, determined by the distance-$i$ graphs ${\mathcal S}_i$ of the Biggs-Smith graph $\mathcal S$, for $i=1,2,3,4$. From this, it is concluded that ${\mathcal Y}={\mathcal S}_3$. In addition, $\mathcal Y$ has each edge (resp. triangle) shared exactly by 4 copies of $L(Q_3)$ (resp. two copies of $L(Q_3)$ plus one of $K_4$).
The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2... more The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ul\-tra\-ho\-mo\-ge\-neous graph $G_3^1$. By natural generalization, $G_3^1$ takes to connected graphs $G_r^\sigma$ ($2<r\in\mathbb{Z}$ and $\sigma\in(0,r-1)\cap\mathbb{Z}$) fitting a definition of $\mathcal C$-{\it ho\-mo\-ge\-neous graph} $G$, with $\mathcal C$ being a class of graphs, that generalizes that of $\mathcal C$-ul\-tra\-ho\-mo\-ge\-neous graph by taking each induced subgraph $Y$ of $G$ in $\mathcal C$ with a distinguished fixed arc. In our case, ${\mathcal C}=\{K_{2s},T_{ts,t}\}$, where $t=2^{\sigma+1}-1$, $s=2^{r-\sigma-1}$ and each edge of $G$ shared by exactly one copy of the complete graph $K_{2s}$ and one of the Turán graph $T_{ts,t}$. Moreover, if $r-\sigma=2$, then $G_r^\sigma$ is $K_4$-ultrahomogeneous with order $(2^r-1)(2^r-2)$, and $2^{\sigma+1}$ edge-disjoint copies of $K_4$ at each vertex.
A construction based on the Biggs-Smith graph is shown to produce an edge-disjoint union of 102 c... more A construction based on the Biggs-Smith graph is shown to produce an edge-disjoint union of 102 copies of $K_4$ forming a $\{K_4,L(Q_3)\}_{K_3}$-ultrahomogeneous graph dressed as the non-line-graphical Menger graph of a self-dual $(102_4)$-configuration. This stands in contrast to the self-dual $(42_4)$-configuration of \cite{D1}, whose Menger graph is a non-line-graphical $\{K_4,$ $K_{2,2,2}\}_{K_2}$-ultrahomogeneous graph.
Let $\mathcal C$ be a class of graphs. A restricted concept of $\mathcal C$-homogeneous graph tha... more Let $\mathcal C$ be a class of graphs. A restricted concept of $\mathcal C$-homogeneous graph that generalizes that of $\mathcal C$-ultrahomogeneous graph in [5] is given, with the explicit construction of $\mathcal C$-homogeneous graphs that are not $\mathcal C$-ultrahomogeneous, more specifically $\{K_{2s},T_{ts,t}\}$-homogeneous graphs. This construction employs ordered pencils of binary projective spaces and includes as an exceptional initial case the $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph of [2].
The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2... more The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ul\-tra\-ho\-mo\-ge\-neous graph $G_3^1$. By natural generalization, $G_3^1$ takes to connected graphs $G_r^\sigma$ ($2&lt;r\in\mathbb{Z}$ and $\sigma\in(0,r-1)\cap\mathbb{Z}$) fitting a definition of $\mathcal C$-{\it ho\-mo\-ge\-neous graph} $G$, with $\mathcal C$ being a class of graphs, that generalizes that of $\mathcal C$-ul\-tra\-ho\-mo\-ge\-neous graph by taking each induced subgraph $Y$ of $G$ in $\mathcal C$ with a distinguished fixed arc. In our case, ${\mathcal C}=\{K_{2s},T_{ts,t}\}$, where $t=2^{\sigma+1}-1$, $s=2^{r-\sigma-1}$ and each edge of $G$ shared by exactly one copy of the complete graph $K_{2s}$ and one of the Turán graph $T_{ts,t}$. Moreover, if $r-\sigma=2$, then $G_r^\sigma$ is $K_4$-ultrahomogeneous with order $(2^r-1)(2^r-2)$, and $2^{\sigma+1}$ edge-disjoint copies of $K_4$ at each vertex.
The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient graph $R_k$ whose vert... more The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient graph $R_k$ whose vertices represent all $k$-edge ordered (rooted) trees $T$. Each $T$ is encoded as a $(2k+1)$-string $F(T)$ that annotates by means of DFS the descending nodes via Kierstead-Trotter lexical colors $0,\ldots,k$, and the ascending edges via $k$ asterisks. This $T$ is assigned a restricted-growth $k$-string in two ways. One of these two ways is obtained via nested substring-swaps in the $F(T)$'s (i.e. pruning-and-regrafting in the $T$'s). These swaps allow sorting the vertices of all $R_k$'s in a canonical, natural, enumerative way that unifies the presentation of the $M_k$'s and their Hamilton cycles by T. Mütze et al.
A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graphs is gi... more A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graphs is given via their dihedral quotients whose vertices are ordered rooted trees.
Let $0<k\in\mathbb{Z}$. A system of numeration according to which the nonnegative integers are pr... more Let $0<k\in\mathbb{Z}$. A system of numeration according to which the nonnegative integers are presented as restricted growth strings has the $k$-th Catalan number $C_k={2k+1\choose k}/(2k+1)$ presented as the restricted growth string $10^k$. This system yields a linear ordering of the vertex set of a quotient of the graph $M_k$ induced by the $k$- and $(k+1)$-levels of the $n$-cube graph under a natural dihedral group action. Mütze proved the existence of Hamilton cycles of $M_k$. We ask for the existence of Hamilton cycles of $M_k$ with dihedral symmetry so the cited system applies to them.
A system of numeration in which every $k$, with $0<k\in\mathbb{Z}$, appears as a restricted growt... more A system of numeration in which every $k$, with $0<k\in\mathbb{Z}$, appears as a restricted growth string, or RGS, has the $k$-th Catalan number as the RGS $10^k$. This induces a canonical ordering of the vertices of the dihedral quotients of the middle-levels graphs.
Let $\mathcal C$ be a class of graphs. A definition of $\mathcal C$-homogeneous graph $G$ is give... more Let $\mathcal C$ be a class of graphs. A definition of $\mathcal C$-homogeneous graph $G$ is given that fulfills the absence of a fitting generalization of $\mathcal C$-ultrahomogeneous (UH) graph, by considering each induced subgraph of $G$ in $\mathcal C$ as anchored through an arc. Let $2<r\in\Z$, $\sigma\in(0,r-1)\cap\Z$, =2^{\sigma+1}-1$ and $s=2^{r-\sigma-1}$. A construction of non-line-graphical $\{K_{2s},T_{ts,t}}_{K_2}$-homogeneous graphs (meaning ${\mathcal C}=\{K_{2s},T_{ts,t}\}$ with each edge of $G$ shared by exactly one copy of the complete subgraph $K_{2s}$ and one copy of the Turán graph $T_{ts,t}$) is conjectured to yield an infinity of such graphs in terms of configurations of points and lines, one per pair $(r,\sigma)$. Such a construction, based on ordered pencils of binary projective spaces, is shown to yield 21 such graphs not $\{K_{2s},T_{ts,t}\}_{K_2}$-UH graphs. For $s=2$, these are shown to be $K_4$-UH graphs, amounting to six of the 21 graphs, for $r=\sigma+2$ and $\sigma=1,...,6$, with orders $(2^r-1)(2^r-2)$, namely 42, 210, 930, 3906, 16002, 64770, and numbers $4(2^{\sigma-1})$ of edge-disjoint copies of $K_4$ incident to each vertex, namely 4, 12, 28, 60, 124, 252, respectively.
A connected edge-disjoint union $\mathcal Y$ of 102 tetrahedra (copies of $K_4$) is constructed w... more A connected edge-disjoint union $\mathcal Y$ of 102 tetrahedra (copies of $K_4$) is constructed which is Menger graph of a self-dual $(102_4)$-configuration and $K_4$-ultrahomogeneous. As $\mathcal Y$ is not a line graph, we ask whether there exists a non-line-graphical $K_4$-ultrahomogeneous Menger graph of a self-dual $(n_4)$-configuration which is connected edge-disjoint union of $n$ copies of $K_4$, for $n\notin\{42,102\}$. Moreover, $\mathcal Y$ is union of 102 cuboctahedra (copies of $L(Q_3)$) with no two sharing a chordless 4-cycle and has an $L(Q_3)$-ultrahomogeneity property restricted to preserving an edge partition of each $L(Q_3)$ into 2-paths, determined by the distance-$i$ graphs ${\mathcal S}_i$ of the Biggs-Smith graph $\mathcal S$, for $i=1,2,3,4$. From this, it is concluded that ${\mathcal Y}={\mathcal S}_3$. In addition, $\mathcal Y$ has each edge (resp. triangle) shared exactly by 4 copies of $L(Q_3)$ (resp. two copies of $L(Q_3)$ plus one of $K_4$).
The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2... more The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ul\-tra\-ho\-mo\-ge\-neous graph $G_3^1$. By natural generalization, $G_3^1$ takes to connected graphs $G_r^\sigma$ ($2<r\in\mathbb{Z}$ and $\sigma\in(0,r-1)\cap\mathbb{Z}$) fitting a definition of $\mathcal C$-{\it ho\-mo\-ge\-neous graph} $G$, with $\mathcal C$ being a class of graphs, that generalizes that of $\mathcal C$-ul\-tra\-ho\-mo\-ge\-neous graph by taking each induced subgraph $Y$ of $G$ in $\mathcal C$ with a distinguished fixed arc. In our case, ${\mathcal C}=\{K_{2s},T_{ts,t}\}$, where $t=2^{\sigma+1}-1$, $s=2^{r-\sigma-1}$ and each edge of $G$ shared by exactly one copy of the complete graph $K_{2s}$ and one of the Turán graph $T_{ts,t}$. Moreover, if $r-\sigma=2$, then $G_r^\sigma$ is $K_4$-ultrahomogeneous with order $(2^r-1)(2^r-2)$, and $2^{\sigma+1}$ edge-disjoint copies of $K_4$ at each vertex.
A construction based on the Biggs-Smith graph is shown to produce an edge-disjoint union of 102 c... more A construction based on the Biggs-Smith graph is shown to produce an edge-disjoint union of 102 copies of $K_4$ forming a $\{K_4,L(Q_3)\}_{K_3}$-ultrahomogeneous graph dressed as the non-line-graphical Menger graph of a self-dual $(102_4)$-configuration. This stands in contrast to the self-dual $(42_4)$-configuration of \cite{D1}, whose Menger graph is a non-line-graphical $\{K_4,$ $K_{2,2,2}\}_{K_2}$-ultrahomogeneous graph.
Let $\mathcal C$ be a class of graphs. A restricted concept of $\mathcal C$-homogeneous graph tha... more Let $\mathcal C$ be a class of graphs. A restricted concept of $\mathcal C$-homogeneous graph that generalizes that of $\mathcal C$-ultrahomogeneous graph in [5] is given, with the explicit construction of $\mathcal C$-homogeneous graphs that are not $\mathcal C$-ultrahomogeneous, more specifically $\{K_{2s},T_{ts,t}\}$-homogeneous graphs. This construction employs ordered pencils of binary projective spaces and includes as an exceptional initial case the $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph of [2].
The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2... more The 42 ordered pencils of the Fano plane are the vertices of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ul\-tra\-ho\-mo\-ge\-neous graph $G_3^1$. By natural generalization, $G_3^1$ takes to connected graphs $G_r^\sigma$ ($2&lt;r\in\mathbb{Z}$ and $\sigma\in(0,r-1)\cap\mathbb{Z}$) fitting a definition of $\mathcal C$-{\it ho\-mo\-ge\-neous graph} $G$, with $\mathcal C$ being a class of graphs, that generalizes that of $\mathcal C$-ul\-tra\-ho\-mo\-ge\-neous graph by taking each induced subgraph $Y$ of $G$ in $\mathcal C$ with a distinguished fixed arc. In our case, ${\mathcal C}=\{K_{2s},T_{ts,t}\}$, where $t=2^{\sigma+1}-1$, $s=2^{r-\sigma-1}$ and each edge of $G$ shared by exactly one copy of the complete graph $K_{2s}$ and one of the Turán graph $T_{ts,t}$. Moreover, if $r-\sigma=2$, then $G_r^\sigma$ is $K_4$-ultrahomogeneous with order $(2^r-1)(2^r-2)$, and $2^{\sigma+1}$ edge-disjoint copies of $K_4$ at each vertex.
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