86 research outputs found

    Proper Hamiltonian Cycles in Edge-Colored Multigraphs

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    A cc-edge-colored multigraph has each edge colored with one of the cc available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.Comment: 13 page

    Thoughts on Barnette's Conjecture

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    We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let GG be a planar triangulation. Then the dual GG^* is a cubic 3-connected planar graph, and GG^* is bipartite if and only if GG is Eulerian. We prove that if the vertices of GG are (improperly) coloured blue and red, such that the blue vertices cover the faces of GG, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then GG^* is Hamiltonian. This result implies the following special case of Barnette's Conjecture: if GG is an Eulerian planar triangulation, whose vertices are properly coloured blue, red and green, such that every red-green cycle contains a vertex of degree 4, then GG^* is Hamiltonian. Our final result highlights the limitations of using a proper colouring of GG as a starting point for proving Barnette's Conjecture. We also explain related results on Barnette's Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published.Comment: 12 pages, 7 figure

    Odd properly colored cycles in edge-colored graphs

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    It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this result be further generalized to more general graphs such as edge-colored graphs? In this paper, we study this problem and show how to decide if there exists an odd properly colored cycle in a given edge-colored graph. As a by-product, we show how to detect if there is a perfect matching in a graph with even (or odd) number of edges in a given edge set

    Uniquely edge colourable graphs

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    On the tractability of some natural packing, covering and partitioning problems

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    In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph G=(V,E)G=(V,E) and two "object types" A\mathrm{A} and B\mathrm{B} chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type A\mathrm{A} and one of type B\mathrm{B} in the edge set EE of GG, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition EE into an object of type A\mathrm{A} and one of type B\mathrm{B}? \textbf{Covering problem:} can we cover EE with an object of type A\mathrm{A}, and an object of type B\mathrm{B}? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an ss-tt path PP and an ss'-tt' path PP' that are edge-disjoint. However, many others were not, for example can we find an ss-tt path PEP\subseteq E and a spanning tree TET\subseteq E that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

    Minimal instances for toric code ground states

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    A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from LU-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square- and triangular lattice, such that the quasi-locality of the toric code hamiltonian becomes a distinctive feature. To this end, a purely geometric procedure to transform a given toric code setup into an LC-equivalent graph state is derived. In combination with an algorithmic computation of LC-equivalent graph states, we find the smallest non-trivial setup on the square lattice to contain 5 plaquettes and 16 qubits; on the triangular lattice the number of plaquettes and qubits is reduced to 4 and 9, respectively.Comment: 14 pages, 11 figure
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