8,439 research outputs found

    Hamilton cycles in bidirected complete graphs

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    Zaslavsky observed that the topics of directed cycles in directed graphs and alternating cycles in edge 2-colored graphs have a common generalization in the study of coherent cycles in bidirected graphs. There are classical theorems by Camion, Harary and Moser, Häggkvist and Manoussakis, and Saad which relate strong connectivity and Hamiltonicity in directed "complete" graphs and edge 2-colored "complete" graphs. We prove two analogues to these theorems for bidirected "complete" signed graphs

    Alternating Hamiltonian cycles in 22-edge-colored multigraphs

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    A path (cycle) in a 22-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in 22-edge-colored multigraphs is an NP\mathcal{NP}-complete problem and it has been studied by several authors. In Bang-Jensen and Gutin's book "Digraphs: Theory, Algorithms and Applications", it is devoted one chapter to survey the last results on this topic. Most results on the existence of alternating Hamiltonian paths and cycles concern on complete and bipartite complete multigraphs and a few ones on multigraphs with high monochromatic degrees or regular monochromatic subgraphs. In this work, we use a different approach imposing local conditions on the multigraphs and it is worthwhile to notice that the class of multigraphs we deal with is much larger than, and includes, complete multigraphs, and we provide a full characterization of this class. Given a 22-edge-colored multigraph GG, we say that GG is 22-M\mathcal{M}-closed (resp. 22-NM\mathcal{NM}-closed)} if for every monochromatic (resp. non-monochromatic) 22-path P=(x1,x2,x3)P=(x_1, x_2, x_3), there exists an edge between x1x_1 and x3x_3. In this work we provide the following characterization: A 22-M\mathcal{M}-closed multigraph has an alternating Hamiltonian cycle if and only if it is color-connected and it has an alternating cycle factor. Furthermore, we construct an infinite family of 22-NM\mathcal{NM}-closed graphs, color-connected, with an alternating cycle factor, and with no alternating Hamiltonian cycle.Comment: 15 pages, 20 figure

    Color degree and alternating cycles in edge-colored graphs

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    AbstractLet G be an edge-colored graph. An alternating cycle of G is a cycle of G in which any two consecutive edges have distinct colors. Let dc(v), the color degree of a vertex v, be defined as the maximum number of edges incident with v that have distinct colors. In this paper, we study color degree conditions for the existence of alternating cycles of prescribed length

    Cones of closed alternating walks and trails

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    Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the \emph{alternating cone}. The integral (respectively, {0,1}\{0,1\}) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to searching for a closed alternating trail through a given edge. The latter problem, called alternating reachability, is solved in a companion paper along with related results.Comment: Minor rephrasing, new pictures, 14 page

    Long properly colored cycles in edge colored complete graphs

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    Let KncK_{n}^{c} denote a complete graph on nn vertices whose edges are colored in an arbitrary way. Let Δmon(Knc)\Delta^{\mathrm{mon}} (K_{n}^{c}) denote the maximum number of edges of the same color incident with a vertex of KncK_{n}^{c}. A properly colored cycle (path) in KncK_{n}^{c} is a cycle (path) in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s (1976) proposed the following conjecture: if Δmon(Knc)<n2\Delta^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor, then KncK_{n}^{c} contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if Δmon(Knc)<n2\Delta^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor, then KncK_{n}^{c} contains a properly colored cycle of length at least n+23+1\lceil \frac{n+2}{3}\rceil+1. In this paper, we improve the bound to n2+2\lceil \frac{n}{2}\rceil + 2.Comment: 8 page
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