Alternating Hamiltonian cycles in 22-edge-colored multigraphs

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.Comment: 15 pages, 20 figure

Properly Colored Notions of Connectivity - A Dynamic Survey

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs

Operations research: from computational biology to sensor network

In this dissertation we discuss the deployment of combinatorial optimization methods for modeling and solve real life problemS, with a particular emphasis to two biological problems arising from a common scenario: the reconstruction of the three-dimensional shape of a biological molecule from Nuclear Magnetic Resonance (NMR) data. The fi rst topic is the 3D assignment pathway problem (APP) for a RNA molecule. We prove that APP is NP-hard, and show a formulation of it based on edge-colored graphs. Taking into account that interactions between consecutive nuclei in the NMR spectrum are diff erent according to the type of residue along the RNA chain, each color in the graph represents a type of interaction. Thus, we can represent the sequence of interactions as the problem of fi nding a longest (hamiltonian) path whose edges follow a given order of colors (i.e., the orderly colored longest path). We introduce three alternative IP formulations of APP obtained with a max flow problem on a directed graph with packing constraints over the partitions, which have been compared among themselves. Since the last two models work on cyclic graphs, for them we proposed an algorithm based on the solution of their relaxation combined with the separation of cycle inequalities in a Branch & Cut scheme. The second topic is the discretizable distance geometry problem (DDGP), which is a formulation on discrete search space of the well-known distance geometry problem (DGP). The DGP consists in seeking the embedding in the space of a undirected graph, given a set of Euclidean distances between certain pairs of vertices. DGP has two important applications: (i) fi nding the three dimensional conformation of a molecule from a subset of interatomic distances, called Molecular Distance Geometry Problem, and (ii) the Sensor Network Localization Problem. We describe a Branch & Prune (BP) algorithm tailored for this problem, and two versions of it solving the DDGP both in protein modeling and in sensor networks localization frameworks. BP is an exact and exhaustive combinatorial algorithm that examines all the valid embeddings of a given weighted graph G=(V,E,d), under the hypothesis of existence of a given order on V. By comparing the two version of BP to well-known algorithms we are able to prove the e fficiency of BP in both contexts, provided that the order imposed on V is maintained