A note on a conjecture on consistent cycles

Let ▫$Gamma$▫ denote a finite digraph and let ▫$G$▫ be a subgroup of its automorphism group. A directed cycle ▫$vec{C}$▫ of▫ $Gamma$▫ is called ▫$G$▫-consistent whenever there is an element of ▫$G$▫ whose restriction to▫ $vec{C}$▫ is the 1-step rotation of ▫$vec{C}$▫. In this short note we provea conjecture on ▫$G$▫-consistent directed cycles stated by Steve Wilson

Classification of Cayley Rose Window Graphs

Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs

On tetravalent half-arc-transitive graphs of girth 5

A subgroup of the automorphism group of a graph \G is said to be {\em half-arc-transitive} on \G if its action on \G is transitive on the vertex set of \G and on the edge set of \G but not on the arc set of \G. Tetravalent graphs of girths $3$ and $4$ admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth $5$. We show that, with two exceptions, all such graphs only have directed $5$-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed $5$-cycles, study some of their graph theoretic properties and prove that the $5$-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth $5$ admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth $5$

Rose window graphs underlying rotary maps

Given natural numbers ▫$n ge 3$▫ and ▫$1 le a$▫, ▫$r le n-1$▫, the rose window graph ▫$R_n(a,r)$▫ is a quartic graph with vertex set ▫${x_i verti in {mathbb Z}_n } cup {y_i verti in {mathbb Z}_n }$▫ and edge set ▫${{x_i, x_{i+1}} verti in {mathbb Z}_n } cup {{y_i, y_{i+1}} verti in {mathbb Z}_n } cup {{x_i, y_i} verti in {mathbb Z}_n} cup {{x_{i+a}, y_i} verti in {mathbb Z}_n }$▫. In this paper rotary maps on rose window graphs are considered. In particular, we answer the question posed in [S. Wilson, Rose window graphs, Ars Math. Contemp. 1 (2008), 7-19. http://amc.imfm.si/index.php/amc/issue/view/5] concerning which of these graphs underlie a rotary map