51. Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groupsŠtefko Miklavič, Primož Šparl, 2012, original scientific article Abstract: In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph ▫$\Gamma$▫ is ▫$n$▫-HC-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton cycle of ▫$\Gamma$▫. Similarly, ▫$\Gamma$▫ is weakly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton path of ▫$\Gamma$▫. Moreover, ▫$\Gamma$▫ is strongly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if for every such path $P$ there is a Hamilton path of ▫$\Gamma$▫ starting with ▫$P$▫. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2-HC-extendable and a complete classification of 3-HC-extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4-HP-extendable. Found in: ključnih besedah Summary of found: ...path (HP) extendability are introduced. A connected graph ▫$\Gamma$▫ is ▫$n$▫-HC-extendable if it contains a... Keywords: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian group Published: 15.10.2013; Views: 1289; Downloads: 79 Full text (0,00 KB) |
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53. Cubic Cayley graphs and snarksKlavdija Kutnar, Ademir Hujdurović, Dragan Marušič, 2012, published scientific conference contribution abstract (invited lecture) Found in: ključnih besedah Summary of found: ...Cayley graph, snark, arc-transitive graph, Cayley map, ... Keywords: Cayley graph, snark, arc-transitive graph, Cayley map Published: 15.10.2013; Views: 1292; Downloads: 48 Full text (0,00 KB) |
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55. Classification of cubic symmetric tetracirculants and pentacirculantsBoštjan Frelih, Klavdija Kutnar, 2013, original scientific article Found in: ključnih besedah Summary of found: ...cubic graph, symmetric, semiregular automorphism, tetracirculant, pentacirculant, ... Keywords: cubic graph, symmetric, semiregular automorphism, tetracirculant, pentacirculant Published: 15.10.2013; Views: 1441; Downloads: 53 Full text (0,00 KB) |
56. Special issue of discrete mathematics on Hamiltonicity problem for vertex-transitive (Cayley) graphsDragan Marušič, 2009, preface, afterword Found in: ključnih besedah Summary of found: ...Hamilton cycle, Hamilton path, Vetrex-transitive graph, Cayley graph, ... Keywords: Hamilton cycle, Hamilton path, Vetrex-transitive graph, Cayley graph Published: 15.10.2013; Views: 1371; Downloads: 14 Full text (0,00 KB) |
57. 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: ključnih besedah Summary of found: ...if Cay▫$(G; S)$▫ is a connected Cayley graph with ▫$n$▫ vertices, and the prime factorization... Keywords: graph theory, Cayley graphs, hamiltonian cycles Published: 15.10.2013; Views: 1652; Downloads: 66 Full text (0,00 KB) |
58. Distance-transitive graphs admit semiregular automorphismsKlavdija 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: ključnih besedah Summary of found: ...A distance-transitive graph is a graph in which for every... Keywords: distance-transitive graph, vertex-transitive graph, semiregular automorphism, permutation group Published: 15.10.2013; Views: 1728; Downloads: 60 Full text (0,00 KB) |
59. Arc-transitive cycle decompositions of tetravalent graphsŠtefko Miklavič, Primož Potočnik, Stephen Wilson, 2008, original scientific article Abstract: A cycle decomposition of a graph ▫$\Gamma$▫ is a set ▫$\mathcal{C}$▫ of cycles of ▫$\Gamma$▫ such that every edge of ▫$\Gamma$▫ belongs to exactly one cycle in ▫$\mathcal{C}$▫. Such a decomposition is called arc-transitive if the group of automorphisms of ▫$\Gamma$▫ that preserve setwise acts transitively on the arcs of ▫$\Gamma$▫. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure. Found in: ključnih besedah Summary of found: ...A cycle decomposition of a graph ▫$\Gamma$▫ is a set ▫$\mathcal{C}$▫ of cycles... Keywords: mathematics, graph theory, cycle decomposition, automorphism group, consistent cycle, medial maps Published: 15.10.2013; Views: 1456; Downloads: 51 Full text (0,00 KB) |
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