Disjoint Cycles Research Articles

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Group Theory and Solutions of the Rubik’s Cube

The paper explores the application of group theory to solving the Rubik’s Cube. Group theory, a branch of abstract algebra, studies sets equipped with an operation that satisfies specific properties: closure, associativity, identity, and inverses. The Rubik’s Cube is treated as a group because its set of moves forms a structure that aligns with these properties. By representing the cube’s various configurations and moves through the mathematical constructs of groups, the paper analyzes how the elements transform during cube manipulations. Key concepts such as symmetric groups, homomorphisms, alternating groups, and disjoint cycle decomposition are used to break down the cube’s complexity. Additionally, the study demonstrates how group actions, particularly the parity of permutations and orientation sums, govern valid cube configurations. The paper also references methodologies from other studies to apply and refine these mathematical approaches for systematically solving the Rubik’s Cube. Through these methods, the research illustrates how the cube can be navigated using logical algorithms grounded in group theory principles

Expressing even permutations as commutators

We prove that an element of the symmetric group of degree n > 3 can be represented as the commutator of two permutations whose supports intersect at exactly p ≤ n points if and only if it is the product of p cycles of length 3. As an accompanying result, and also as a step of the main result’s proof, we give a lower bound for p to express an even permutation as the product of p cycles of length 3, in terms of its decomposition into disjoint cycles.

On the Restricted Steiner Multi Cycle Problem

We introduce a new problem, called Steiner Multi Cycle Problem, in which we are given a complete weighted graph G=(V,E), which respects the triangle inequality, a collection of terminal sets {T1,…,Tk}, where for each a in [k] we have a subset Ta of V and these terminal sets are pairwise disjoint. The goal is to find a set of disjoint cycles of minimum cost such that for each a in [k], all vertices of Ta belong to a same cycle. Our main interest is in a restricted case where |Ta|=2, for each a in [k], which models a collaborative problem with pickup and delivery. We show that even the restricted problem is NP-Hard, and present three heuristics to solve it: a dynamic programming algorithm called Refinement Search Heuristic, which explores geometric and laminar set properties; a heuristic based on the Gonzales Clustering algorithm; and a GRASP Heuristic with Path Relinking. We performed computational experiments with 525 instances to compare these methods, which achieved results close to the optimum, highlighting the performance of the Refinement Search Heuristic both in quality and time.

Cycle‐factors in oriented graphs

AbstractLet be a positive integer. A ‐cycle‐factor of an oriented graph is a set of disjoint cycles of length that covers all vertices of the graph. In this paper, we prove that there exists a positive constant such that for sufficiently large, any oriented graph on vertices with both minimum out‐degree and minimum in‐degree at least contains a ‐cycle‐factor for any . Additionally, under the same hypotheses, we also show that for any sequence with and the number of the equal to 3 is , where is any real number with , the oriented graph contains disjoint cycles of lengths . This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.

Injectives over Leavitt path algebras of graphs with disjoint cycles

Let K be any field, and let E be a finite graph with the property that every vertex in E is the base of at most one cycle (i.e., a graph with disjoint cycles). We explicitly construct the injective envelope of each simple left module over the Leavitt path algebra LK(E). The main idea girding our construction is that of a “formal power series” extension of modules, thereby developing for all graphs with disjoint cycles the understanding of injective envelopes of simple modules over LK(E) achieved previously for the simple modules over the Toeplitz algebra.

Root cycles in Coxeter groups

Abstract For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

A solution to the conjecture of vertex disjoint cycles in graphs with partial degree

A solution to the conjecture of vertex disjoint cycles in graphs with partial degree

Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks

This research delves into the extensive analysis of symmetric groups of various orders and degrees. We explore different representations of permutations, construct a Cayley table for the symmetric group of degree four, and systematically identify all its subgroups using Lagrange's theorem. An intriguing discovery unfolds as we demonstrate that the converse of the theorem countered by Sylow's theorem does not hold universally. Furthermore, we apply the concepts of permutations and product of disjoint cycles to address a real-world problem – the Card Trick game. This study amalgamates theoretical exploration with practical applications, showcasing the versatility and depth of symmetric group theory.

Embedding Spanning Disjoint Cycles in Hypercube Networks with Prescribed Edges in Each Cycle

One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,…,ek of G, there exist k vertex-disjoint cycles C1,C2,…,Ck in G such that V(C1)∪V(C2)∪⋯∪V(Ck)=V(G) and ei∈E(Ci) for all 1≤i≤k. According to the definition, the problem of finding hamiltonian cycle focuses on k=1. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Qn is spanning k-edge-cyclable for 1≤k≤n−1 and n≥2. This is the best possible result, in the sense that the n-dimensional hypercube Qn is not spanning n-edge-cyclable.

Embedding spanning disjoint cycles in augmented cube networks with prescribed vertices in each cycle

One of the important issues in evaluating an interconnection network is to study the Hamiltonian cycle embedding problems. For a positive integer k, a graph G is said to be spanning k-cyclable if for k prescribed vertices , there exist k disjoint cycles such that the union of spans G, and each contains exactly one vertex of . According to the definition, the problem of finding hamiltonian cycle focuses on k = 1. The notion of spanning cyclability can be applied to the problem of identifying faulty processors and other related issues in interconnection networks. The n-dimensional augmented cube is an important node-symmetric variant of the n-dimensional hypercube . In this paper, we prove that with is spanning k-cyclable for .

On the existence of k $k$‐cycle semiframes for even k $k$

AbstractA ‐semiframe of type is a ‐group divisible design of type in which is the vertex set, is the group set, and the set of ‐cycles can be written as a disjoint union where is partitioned into parallel classes on and is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition or for some . In this paper, we almost completely solve the existence of a ‐semiframe of type for all and a ‐semiframe of type for all and .

A new class of higher quantum Airy structures as modules of $\mathcal{W}(\mathfrak{gl}_r)$-algebras

Quantum rr-Airy structures can be constructed as modules of \mathcal{W}(\mathfrak{gl}_r)𝒲(𝔤𝔩r)-algebras via restriction of twisted modules for the underlying Heisenberg algebra. In this paper we classify all such higher quantum Airy structures that arise from modules twisted by automorphisms of the Cartan subalgebra consisting of products of disjoint cycles of the same length. An interesting feature of these higher quantum Airy structures is that the dilaton shifts must be chosen carefully to satisfy a matrix invertibility condition, with a natural choice being roots of unity. We explore how these higher quantum Airy structures may provide a definition of the Chekhov, Eynard, and Orantin topological recursion for reducible algebraic spectral curves. We also study under which conditions quantum rr-Airy structures that come from modules twisted by arbitrary automorphisms can be extended to new quantum (r+1)(r+1)-Airy structures by appending a trivial one-cycle to the twist without changing the dilaton shifts.

Rooted Prism-Minors and Disjoint Cycles Containing a Specified Edge

Dirac and Lovász independently characterized the $3$-connected graphs with no pair of vertex-disjoint cycles. Equivalently, they characterized all $3$-connected graphs with no prism-minors. In this paper, we completely characterize the $3$-connected graphs with no edge that is contained in the union of a pair of vertex-disjoint cycles. As applications, we answer the analogous questions for edge-disjoint cycles and for $4$-connected graphs and we completely characterize the $3$-connected graphs with no prism-minor using a specified edge.

Linking number of monotonic cycles in random book embeddings of complete graphs

A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining “interior” edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of graphs. In particular, we show that the distribution of linking numbers of disjoint monotonic cycles can be described in terms of Eulerian numbers. Consequently, the mean of the squared linking number over all random embeddings is [Formula: see text], where [Formula: see text] is the number of interior edges in the cycles. We also show that the mean of the squared linking number over all pairs of disjoint monotonic [Formula: see text]-cycles in [Formula: see text] grows linearly in [Formula: see text].

Two disjoint cycles in digraphs

AbstractBermond and Thomassen conjectured that every digraph with minimum outdegree at least contains vertex disjoint cycles. So far the conjecture was verified for . Here we generalise the question asking for all outdegree sequences which force vertex disjoint cycles and give the full answer for .

Disjoint Cycles and Degree Sum Condition in a Graph

Disjoint Cycles and Degree Sum Condition in a Graph

Nominal Sets in Agda - A Fresh and Immature Mechanization

In this paper we present our current development on a new formalization of nominal sets in Agda. Our first motivation in having another formalization was to understand better nominal sets and to have a playground for testing type systems based on nominal logic. Not surprisingly, we have independently built up the same hierarchy of types leading to nominal sets. We diverge from other formalizations in how to conceive finite permutations: in our formalization a finite permutation is a permutation (i.e. a bijection) whose domain is finite. Finite permutations have different representations, for instance as compositions of transpositions (the predominant in other formalizations) or compositions of disjoint cycles. We prove that these representations are equivalent and use them to normalize (up to composition order of independent transpositions) compositions of transpositions.

Partition and Disjoint Cycles in Digraphs

Partition and Disjoint Cycles in Digraphs

Disjoint Cycles in a Digraph with Partial Degree

Disjoint Cycles in a Digraph with Partial Degree