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A corner in a map is an edge-vertex-edge triple consisting of two distinct edges incident to the same vertex. A corneration is a set of corners that covers every arc of the map exactly once. Cornerations in a dart-transitive map generalize the notion of a cycle structure in a symmetric graph. In this paper, we study the cornerations (and associated structures) that are preserved by a vertex-transitive group of automorphisms of the map.

A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length. While the tetravalent edge-transitive graphs admitting a semiregular automorphism with only one orbit are easy to determine, those that admit a semiregular automorphism with two orbits took a considerable effort and were finally classified in 2012. Of the several possible different ``types'' of potential tetravalent edge-transitive graphs admitting a semiregular automorphism with three orbits, only one ``type'' has thus far received no attention. In this paper we focus on this class of graphs, which we call the Woolly Hat graphs. We prove that there are in fact no edge-transitive Woolly Hat graphs and classify the vertex-transitive ones., Comment: 26 pages

A bipartite biregular $(n,m;g)$-graph $G$ is a bipartite graph of even girth $g$ having the degree set $\{n,m\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An $(n,m;g)$-bipartite biregular cage is a bipartite biregular $(n,m;g)$-graph of minimum order. In their 2019 paper, Filipovski, Ramos-Rivera and Jajcay present lower bounds on the orders of bipartite biregular $(n,m;g)$-graphs, and call the graphs that attain these bounds {\em bipartite biregular Moore cages}. In parallel with the well-known classical results relating the existence of $k$-regular Moore graphs of even girths $g = 6,8 $ and $12$ to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of $S(2,k,v)$-Steiner systems yields the existence of bipartite biregular $(k,\frac{v-1}{k-1};6)$-Moore cages. Moreover, in the special case of Steiner triple systems (i.e., in the case $k=3$), we completely solve the problem of the existence of $(3,m;6)$-bipartite biregular cages for all integers $m\geq 4$. Considering girths higher than $6$ and prime powers $s$, we relate the existence of generalized polygons (quadrangles, hexagons and octagons) with the existence of $(n+1,n^2+1;8)$, $(n+1,n^3+1;12)$, and $(n+1,n^2+1;16)$-bipartite biregular Moore cages, respectively. Using this connection, we derive improved upper bounds for the orders of bipartite biregular cages of girths $8$, $12$ and $14$., Comment: 12 pages

Let ${\cal M}$ be a map with the underlying graph $\Gamma$. The automorphism group $Aut({\cal M})$ induces a natural action on the set of all vertex-edge-face incident triples, called {\em flags} of ${\cal M}$. The map ${\cal M}$ is said to be a {\em $k$-orbit} map if $Aut({\cal M})$ has $k$ orbits on the set of all flags of ${\cal M}$. It is known that there are seven different classes of $2$-orbit maps, with only four of them corresponding to arc-transitive maps, that is maps for which $Aut{\cal M}$ acts arc-transitively on the underlying graph $\Gamma$. The Petrie dual operator links these four classes in two pairs, one of which corresponds to the chiral maps and their Petrie duals. In this paper we focus on the other pair of classes of $2$-orbit arc-transitive maps. We investigate the connection of these maps to consistent cycles of the underlying graph with special emphasis on such maps of smallest possible valence, namely $4$. We then give a complete classification of such maps whose underlying graphs are arc-transitive Rose Window graphs.