Alejandro Contreras-Balbuena ; Hortensia Galeana-Sánchez ; Ilan A. Goldfeder - Alternating Hamiltonian cycles in $2$-edge-colored multigraphs

dmtcs:4882 - Discrete Mathematics & Theoretical Computer Science, June 13, 2019, vol. 21 no. 1, ICGT 2018 - https://doi.org/10.23638/DMTCS-21-1-12
Alternating Hamiltonian cycles in $2$-edge-colored multigraphsArticle

Authors: Alejandro Contreras-Balbuena ; Hortensia Galeana-Sánchez ; Ilan A. Goldfeder

    A path (cycle) in a $2$-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 $2$-edge-colored multigraphs is an $\mathcal{NP}$-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 $2$-edge-colored multigraph $G$, we say that $G$ is $2$-$\mathcal{M}$-closed (resp. $2$-$\mathcal{NM}$-closed)} if for every monochromatic (resp. non-monochromatic) $2$-path $P=(x_1, x_2, x_3)$, there exists an edge between $x_1$ and $x_3$. In this work we provide the following characterization: A $2$-$\mathcal{M}$-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 $2$-$\mathcal{NM}$-closed graphs, color-connected, with an alternating cycle factor, and with no alternating Hamiltonian cycle.


    Volume: vol. 21 no. 1, ICGT 2018
    Published on: June 13, 2019
    Accepted on: May 28, 2019
    Submitted on: October 11, 2018
    Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics,05C15, 05C45

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