1. Primitive, edge-short, isometric, and pantochordal cyclesGover E. C. Guzman, Marcos E. González Laffitte, André Fujita, Peter F. Stadler, 2025, original scientific article Abstract: A cycle in a graph G is said to be primitive from its vertex x if at least one of its edges does not belong to any shorter cycle that passes through x. This type of cycle and an associated notion of extended neighborhoods play a key role in message-passing algorithms that compute spectral properties of graphs with short loops. Here, we investigate such primitive cycles and graphs without long primitive cycles in a more traditional graph-theoretic framework. We show that a cycle is primitive from all its vertices if and only if it is isometric. We call a cycle fully redundant cycles if it is not primitive from any of its vertices and show that fully redundant cycles, in particular, are not edge short, i.e., they cannot be represented as the edge-disjoint union of a single edge and two shortest paths in G. The families Rk and Lk of graphs with all cycles of length at least k + 1 being fully redundant and not edge-short, respectively, coincide for k = 3 and k = 4. In these graphs, all cycles of length at least k + 1 are pantochordal, i.e., each of their vertices is incident with a chord. None of these results generalizes to k ≥ 5. Moreover, R₃ = L₃ turn out to be the block graphs, and R₄ = L₄ are the graphs with complete multi-partite blocks. The cographs, finally, are shown to form a proper subset of R₅.
Keywords: edge-short cycle, chord, block-graph, complete multipartite graph, wheel graphs, cographs, geodesic cycles, Hamiltonian cycles
Published in RUP: 03.11.2025; Views: 121; Downloads: 0
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2. 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: 2142; Downloads: 19
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3. 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: 5349; Downloads: 129
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