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1.
Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms
Dragan Marušič, 2017, original scientific article

Abstract: It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.
Keywords: solvable group, semiregular automorphism, fixed-point-free automorphism, polycirculant conjecture
Published in RUP: 03.01.2022; Views: 763; Downloads: 17
.pdf Full text (235,26 KB)

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Bicirculants via imprimitivity block systems
Dragan Marušič, 2021, original scientific article

Keywords: semiregular group, imprimitive group, arc-transitive graph, bicirculant
Published in RUP: 06.05.2021; Views: 784; Downloads: 30
URL Link to full text

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Hamilton cycles in primitive vertex-transitive graphs of order a product of two primes - the case PSL(2, q[sup]2) acting on cosets of PGL(2, q)
Shao Fei Du, Klavdija Kutnar, Dragan Marušič, 2020, original scientific article

Abstract: A step forward is made in a long standing Lovász problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes arising from the group action of the projective special linear group PSL▫$(2, q^2)$▫ on cosets of its subgroup isomorphic to the projective general linear group PGL$(2, q)$ contains a Hamilton cycle.
Keywords: vertex-transitive graph, Hamilton cycle, automorphism group, orbital graph
Published in RUP: 20.07.2020; Views: 1247; Downloads: 46
.pdf Full text (365,31 KB)

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Polynomials of degree 4 over finite fields representing quadratic residues
Shao Fei Du, Klavdija Kutnar, Dragan Marušič, 2019, original scientific article

Keywords: finite field, polynomial, quadratic residues
Published in RUP: 11.02.2020; Views: 1228; Downloads: 74
.pdf Full text (317,71 KB)

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Lovász Hamiltonicity Problem
Klavdija Kutnar, Shao Fei Du, Dragan Marušič, 2019, published scientific conference contribution abstract (invited lecture)

Keywords: Lovász problem, Hamilton cycle, vertex-transitive graph
Published in RUP: 06.08.2019; Views: 1626; Downloads: 87
.pdf Full text (19,29 KB)
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