Cycles and symmetries in graphsLekše, Maruša (Avtor) Potočnik, Primož (Mentor) Toledo Roy, Micael Alexi (Komentor) Vertex-transitive graphsarc-transitive graphsgirth-regularityrotary mapsconsistent cyclesOberwolfach problemsymmetry type graph conjectureThis thesis explores the interplay between cycles and symmetry in finite graphs, studying several sets of cycles that are invariant under the action of a group of automorphisms (girth cycles, face cycles and consistent cycles), and the constraints they impose on the graph and its automorphism group. We classify cubic vertex-transitive graphs of girth $7$, extending the existing classifications for girth up to $6$ due to Potočnik and Vidali and to Eiben, Šparl and Jajcay. Additionally, we observe that every vertex-transitive map of type $\{7,3\}$ is rotary. This observation motivates a broader question: under what conditions on the signature does every vertex-transitive group of automorphisms of every map of that signature act as a rotary group? We give a complete characterization: every map of type $\{p,q\}$ has the property that every vertex-transitive group of automorphisms is rotary if and only if $q$ is odd and ${\rm gcd}(p,q!) = 1$. Next we study consistent cycles in arc-transitive graphs. A cycle in a graph is $G$-consistent if some automorphism in $G$ cyclically permutes its vertices by one step. We give sufficient conditions for the existence of a $G$-consistent walk with trivial pointwise stabilizer. These conditions hold whenever the local group is weakly $p$-subregular, and also whenever the graph has valence at most $4$. As an application, we show that every vertex-transitive group of automorphisms of a connected $d$-valent graph can always be generated by the stabilizer of any consistent walk, plus at most $d$ automorphisms. We address the spouse-loving variant of the Oberwolfach problem. We develop techniques for transforming solutions of ${\rm OP}$ into solutions of ${\rm OP}^+$, and use them to completely resolve the two-table case: ${\rm OP}^+(\ell_1, \ell_2)$ has a solution if and only if $\ell_1 + \ell_2$ is even and $(\ell_1, \ell_2)$ is not $(2, 2)$ or $(3, 3)$. Finally, we address a question from the theory of maniplexes on symmetry type graphs. Can every map be obtained as the quotient of some map by its full automorphism group?20262026-10-01 08:15:10Doktorsko delo/naloga189056UDK: 519.17VisID: 165815COBISS_ID: 293282051sl