1. The core of a vertex-transitive complementary prismMarko Orel, 2023, izvirni znanstveni članek Povzetek: The complementary prism $\Gamma \overline{\Gamma}$ is obtained from the union of a graph $\Gamma$ and its complement $\overline{\Gamma}$ where each pair of identical vertices in $\Gamma$ and $\overline{\Gamma}$ is joined by an edge. It generalizes the Petersen graph, which is the complementary prism of the pentagon. The core of a vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-transitive complementary prism $\Gamma \overline{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is a core or its core is a complete graph.
Ključne besede: graph homomorphism, complementary prism, self-complementary graph, vertex-transitive graph, core
Objavljeno v DiRROS: 09.04.2024; Ogledov: 57; Prenosov: 32
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3. The core of a vertex transitive complementary prism of a lexicographic productMarko Orel, 2023, izvirni znanstveni članek Povzetek: The complementary prism of a graph $\Gamma$ is the graph $\Gamma \overline{\Gamma}$, which is formed from the union of $\Gamma$ and its complement $\overline{\Gamma}$ by adding an edge between each pair of identical vertices in $\Gamma$ and $\overline{\Gamma}$. Vertex-transitive self-complementary graphs provide vertex-transitive complementary prisms. It was recently proved by the author that $\Gamma \overline{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is vertex-transitive, self-complementary, and either $\Gamma$ is a core or its core is a complete graph. In this paper the same conclusion is obtained for some other classes of vertex-transitive self-complementary graphs that can be decomposed as a lexicographic product $\Gamma = \Gamma_1 [\Gamma_2]$. In the process some new results aboutthe homomorphisms of a lexicographic product are obtained.
Ključne besede: graph homomorphism, core, complementary prism, self-complementary graph, vertex-transitive graph, lexicographic product
Objavljeno v DiRROS: 19.03.2024; Ogledov: 71; Prenosov: 42
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4. Cubic vertex-transitive graphs admitting automorphisms of large orderPrimož Potočnik, Micael Toledo, 2023, izvirni znanstveni članek Povzetek: A connected graph of order $n$ admitting a semiregular automorphism of order $n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper, we study the broader class of cubic vertex-transitive graphs of order $n$ admitting an automorphism of order $n/3$ or larger that may not be semiregular. In particular, we show that any such graph is either a $k$-multicirculant for some $k \le 3$, or it belongs to an infinite family of graphs of girth $6$.
Ključne besede: cubic vertex-transitive graphs, multicirculants, automorphisms of large order
Objavljeno v DiRROS: 18.03.2024; Ogledov: 80; Prenosov: 44
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5. On orders of automorphisms of vertex-transitive graphsPrimož Potočnik, Micael Toledo, Gabriel Verret, 2024, izvirni znanstveni članek Povzetek: In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $\Gamma$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ belong to a regular orbit of $g$. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms $C$ of a connected $3$-valent vertex-transitive graph $\Gamma$ in terms of the number of vertices of $\Gamma$ and the length of a longest orbit of $C$.
Ključne besede: graphs, automorphism groups, vertex-transitive, regular orbit, cubic, tetravalent
Objavljeno v DiRROS: 19.02.2024; Ogledov: 166; Prenosov: 55
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