317 research outputs found
On rainbow tetrahedra in Cayley graphs
Let be the complete undirected Cayley graph of the odd cyclic
group . Connected graphs whose vertices are rainbow tetrahedra in
are studied, with any two such vertices adjacent if and only if they
share (as tetrahedra) precisely two distinct triangles. This yields graphs
of largest degree 6, asymptotic diameter and almost all vertices
with degree: {\bf(a)} 6 in ; {\bf(b)} 4 in exactly six connected subgraphs
of the -semi-regular tessellation; and {\bf(c)} 3 in exactly four
connected subgraphs of the -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 () has a dihedral quotient
pseudograph whose vertices are the -edge ordered trees , each
encoded as a -string formed via DFS by: {\bf(i)}
(BFS-assigned) Kierstead-Trotter lexical colors for
the descending nodes; {\bf(ii)} asterisks for the ascending edges. Two
ways of corresponding a restricted-growth -string to each
exist, namely one Stanley's way and a novel way that assigns to
via nested substring-swaps. These swaps permit to sort as an ordered
tree that allows a lexical visualization of as well as the Hamilton
cycles of constructed by P. Gregor, T. M\"utze and J. Nummenpalo.Comment: 26 pages, 8 figures, 10 table
On a -ultrahomogeneous oriented graph
The notion of a -ultrahomogeneous graph, due to Isaksen et al.,
is adapted for digraphs, and subsequently a strongly connected
-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
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