1. A novel characterization of cubic Hamiltonian graphs via the associated quartic graphsSimona Bonvicini, Tomaž Pisanski, 2017, original scientific article Abstract: We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian ▫$I$▫-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian ▫$I$▫-graphs follows from the fact that one can choose a 1-factor in any ▫$I$▫-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected ▫$I$▫-graphs as a subfamily. Keywords: generalized Petersen graphs, I-graphs, Hamiltonian cycles, Eulerian tours, Cayley multigraphs Published in RUP: 03.01.2022; Views: 800; Downloads: 16 Full text (1,01 MB) |
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3. Določeni razredi (hiper)grafov in njihove algebraične lastnosti : doktorska disertacijaPaweł Petecki, 2016, doctoral dissertation Keywords: hypergraph, hamiltonian cycle, decomposition, double generalized Petersen graph, automorphism group, vertex-transitive, sign graph, L-eigenvalue, lollipop graph Published in RUP: 09.08.2016; Views: 3179; Downloads: 30 Link to full text |
4. 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 ▫$k \le 2$▫. Keywords: graph theory, Cayley graphs, hamiltonian cycles Published in RUP: 15.10.2013; Views: 3528; Downloads: 121 Full text (545,91 KB) |
5. The spanning laceability on the faulty bipartite hypercube-like networksCheng-Kuan Lin, Yuan-Hsiang Teng, Jimmy J. M. Tan, Lih-Hsing Hsu, Dragan Marušič, 2013, original scientific article Keywords: Hamiltonian, Hamiltonian laceable, Hypercube networks, Hypercube-like network, Spanning laceability Published in RUP: 15.10.2013; Views: 2721; Downloads: 85 Link to full text |