31. Hamilton cycles in (2, odd, 3)Cayley graphsHenry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific article Abstract: In 1969, Lovász asked if every finite, connected vertextransitive 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, arctransitive graph, 1regular action, automorphism group Published: 15.10.2013; Views: 1279; Downloads: 63 Full text (0,00 KB) 
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36. Hamiltonian cycles in Cayley graphs whose order has few prime factorsKlavdija 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 Full text (0,00 KB) 
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39. A note on a geometric construction of large Cayley graps of given degree and diameterGyörgy 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 Full text (0,00 KB) This document has more files! More...

40. Fibonaccijeva števila in zlati rez v umetnostiKatarina 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 Full text (0,00 KB) This document has more files! More...
