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31.
Hamilton cycles in (2, odd, 3)-Cayley graphs
Henry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific article

Abstract: In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.
Found in: osebi
Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group
Published: 15.10.2013; Views: 1279; Downloads: 63
URL Full text (0,00 KB)

32.
On cubic non-Cayley vertex-transitive graphs
Klavdija Kutnar, 2010, invited lecture at foreign university

Found in: osebi
Keywords: non-Cayley, vertex-transitive, automorphism grup
Published: 15.10.2013; Views: 1563; Downloads: 48
URL Full text (0,00 KB)

33.
Cubic Cayley graphs and snarks
Klavdija Kutnar, Ademir Hujdurović, Dragan Marušič, 2012, published scientific conference contribution abstract (invited lecture)

Found in: osebi
Keywords: Cayley graph, snark, arc-transitive graph, Cayley map
Published: 15.10.2013; Views: 1293; Downloads: 48
URL Full text (0,00 KB)

34.
Classification of cubic symmetric tetracirculants and pentacirculants
Boštjan Frelih, Klavdija Kutnar, 2013, original scientific article

Found in: osebi
Keywords: cubic graph, symmetric, semiregular automorphism, tetracirculant, pentacirculant
Published: 15.10.2013; Views: 1448; Downloads: 53
URL Full text (0,00 KB)

35.
Bled'11
Klavdija Kutnar, Primož Šparl, 2013, preface, afterword

Found in: osebi
Published: 15.10.2013; Views: 1049; Downloads: 23
URL Full text (0,00 KB)

36.
Hamiltonian cycles in Cayley graphs whose order has few prime factors
Klavdija Kutnar, Dragan Marušič, D. W. Morris, Joy Morris, Primož Šparl, 2012, original scientific article

Abstract: We prove that if Cay▫$(G; S)$▫ is a connected Cayley graph with ▫$n$▫ vertices, and the prime factorization of ▫$n$▫ is very small, then Cay▫$(G; S)$▫ has a hamiltonian cycle. More precisely, if ▫$p$▫, ▫$q$▫, and ▫$r$▫ are distinct primes, then ▫$n$▫ can be of the form kp with ▫$24 \ne k < 32$▫, or of the form ▫$kpq$▫ with ▫$k \le 5$▫, or of the form ▫$pqr$▫, or of the form ▫$kp^2$▫ with ▫$k \le 4$▫, or of the form ▫$kp^3$▫ with ▫$k \le 2$▫.
Found in: osebi
Keywords: graph theory, Cayley graphs, hamiltonian cycles
Published: 15.10.2013; Views: 1653; Downloads: 66
URL Full text (0,00 KB)

37.
Distance-transitive graphs admit semiregular automorphisms
Klavdija Kutnar, Primož Šparl, 2010, original scientific article

Abstract: A distance-transitive graph is a graph in which for every two ordered pairs ofvertices ▫$(u,v)$▫ and ▫$(u',v')$▫ such that the distance between ▫$u$▫ and ▫$v$▫ is equal to the distance between ▫$u'$▫ and ▫$v'$▫ there exists an automorphism of the graph mapping ▫$u$▫ to ▫$u'$▫ and ▫$v$▫ to ▫$v'$▫. A semiregular element of a permutation group is anon-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism.
Found in: osebi
Keywords: distance-transitive graph, vertex-transitive graph, semiregular automorphism, permutation group
Published: 15.10.2013; Views: 1729; Downloads: 60
URL Full text (0,00 KB)

38.
A note on a geometric construction of large Cayley graps of given degree and diameter
György Kiss, István Kovács, Klavdija Kutnar, János Ruff, Primož Šparl, 2008, published scientific conference contribution abstract

Found in: osebi
Keywords: diameter-degree problem, Cayley graph, projective space
Published: 15.10.2013; Views: 1785; Downloads: 12
URL Full text (0,00 KB)

39.
A note on a geometric construction of large Cayley graps of given degree and diameter
György Kiss, István Kovács, Klavdija Kutnar, János Ruff, Primož Šparl, 2009, original scientific article

Abstract: An infinite series and some sporadic examples of large Cayley graphs with given degree and diameter are constructed. The graphs arise from arcs, caps and other objects of finite projective spaces.
Found in: osebi
Keywords: degree, diameter problem, Moore bound, finite projective spaces
Published: 15.10.2013; Views: 1671; Downloads: 24
URL Full text (0,00 KB)
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40.
Fibonaccijeva števila in zlati rez v umetnosti
Katarina Biščak, 2013, master's thesis

Found in: osebi
Keywords: Fibonaccijeva števila, Fibonaccijevo zaporedje, posplošeno Fibonaccijevo zaporedje, zlati rez, geometrija, likovna umetnost, kiparstvo, arhitektura, glasba
Published: 10.07.2015; Views: 1493; Downloads: 30
URL Full text (0,00 KB)
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