Spanning Edge Cyclability of Enhanced Hypercube Networks

Packing cycles through prescribed vertices

Let k be a positive integer. Let G be a graph of order $$n\ge 3$$n?3 and W a subset of V(G) with $$|W|\ge 3k$$|W|?3k. Wang (J Graph Theory 78:295---304, 2015) proved that if $$d(x)\ge 2n/3$$d(x)?2n/3 for each $$x\in W$$x?W, then G contains k vertex-disjoint cycles such that each of them contains at least three vertices of W. In this paper, we obtain an analogue result of Wang's Theorem in bipartite graph with the partial degree condition. Let $$G=(V_1,V_2;E)$$G=(V1,V2?E) be a bipartite graph with $$|V_1|=|V_2|=n$$|V1|=|V2|=n, and let W be a subset of $$V_1$$V1 with $$|W|\ge 2k$$|W|?2k, where k is a positive integer. We show that if $$d(x)+d(y)\ge n+k$$d(x)+d(y)?n+k for every pair of nonadjacent vertices $$x\in W, y\in V_2$$x?W,y?V2, then G contains k vertex-disjoint cycles such that each of them contains at least two vertices of W.

In [5], we found an upper bound on the number of edges, $\mathcal{E}(G)$, of a graph $G$ containing no $r$ vertex-disjoint cycles of length 3. In this paper we generalize this result to graphs containing no $r$ vertex-disjoint cycles of length $2k+1$. We showed that $\mathcal{E}(G)\les \lfloor \frac{(n-r+1)^2}{4} \rfloor +(r-1)(n-r+1)$ for every $G\in\mathcal{G}(n,V_{r,2k+1})$, the class of all graphs on $n$ vertices containing no $r$ vertex-disjoint cycles of length $2k+1$. Determination of the maximum number of edges in a given graph that contains no specific subgraphs is one of the important problems in graph theory. Solving such problems has attracted the attention of many researchers in graph theory.

Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is the number of independent vertex subsets. In this paper, introduce the identities of domination, independence and Fibonacci numbers of graphs containing vertex-disjoint cycles and edge-disjoint cycles.

The well-known Erdos-Posa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c·k log k (for some constant c) such that G - X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science.In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdos-Posa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way.In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem.We give the first fixed parameter algorithms for the two problems. Namely;1. For fixed k, we can either find a vertex set X of size k such that G -- X has no S-cycle, or conclude that such a vertex set does not exist in O(n2m) time (independently obtained in [7]).2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n2m) time.We also extend the above results to those with the parity constraints as follows;1. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G -- X has no evenS-cycle, or concludes that such a vertex set does not exist.2. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G -- X has no odd S-cycle, or concludes that such a vertex set does not exist.3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint evenS-cycles, or concludes that such k disjoint cycles do not exist.4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.

The Cycle Packing problem asks whether a given undirected graph G=(V,E) contains k vertex-disjoint cycles. Since the publication of the classic Erdos-Posa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson–Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time 2^{O(k^2)}|V| using exponential space. In case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a 2^{O(k\log^2 k)}|V|-time (deterministic) algorithm using exponential space, which is a consequence of the Erdos-Posa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time 2^{o(k\log^2k)}|V|^{O(1)}, beating the bound 2^{O(k\log^2k)}\cdot |V|^{O(1)}, has been found. In light of this, it seems natural to ask whether the 2^{O(k\log^2k)}|V|^{O(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a 2^{O(k\log^2k/log log k})} |V|-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the known bound, our algorithm runs in time linear in |V|, and its space complexity is polynomial in the input size.

Toughness, hamiltonicity and spectral radius in graphs

A Hamiltonian cycle in a graph G is a cycle which contains every vertex of G. The study of Hamiltonian cycle problem has a long history in graph theory and is a central theme. In general, it is NP-complete to decide whether a graph contains a Hamiltonian cycle. Thus researchers have been investigating sufficient conditions that guarantee the existence of a Hamiltonian cycle in a graph. There are many classic results along this line. For example, in 1952, Dirac showed that an n-vertex graph G with n ≥ 3 is Hamiltonian if δ(G) ≥ n. Chv´atal studied Hamiltonian cycles by considering graph toughness, a measure of resilience under the removal of vertices. Let t ≥ 0 be a real number and denote by c(G) the number of components of G. We say a graph G is t-tough if for each cut set S of G we have t · c(G − S) ≤ |S|. The toughness of a graph G, denoted τ (G), is the maximum value of t for which G is t-tough if G is non-complete, and is defined to be ∞ if G is complete. Chv´atal conjectured in 1973 the existence of some constant t such that all t-tough graphs with at least three vertices are Hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in general. Supporting this conjecture of Chv´atal’s, in the first part of this thesis, we show that every 3-tough (P2 ∪ 3P1)-free graph with at least three vertices is Hamiltonian, where P2 ∪ 3P1 is the disjoint union of an edge and three isolated vertices. The notion of a 2-factor is a generalization of a Hamiltonian cycle, which consists of vertex disjoint cycles which together cover the vertices of G. Thus, a Hamiltonian cycle is just a 2-factor with exactly one cycle. It is known that every 2-tough graph with at least three vertices has a 2-factor. In graphs with restricted structures the toughness bound 2 can be improved. For example, it was shown that every 2K2-free 3/2-tough graph with at least three vertices has a 2-factor, and the toughness bound 3/2 is best possible. In viewing 2K2, the disjoint union of two edges, as a linear forest, in this thesis, for any linear forest R on 5, 6, or 7 vertices, we find the sharp toughness bound t such that every t-tough R-free graph

Graph theory is an important topic in discrete mathematics. It is particularly interesting because it has a wide range of applications. Among the main problems in graph theory, we shall mention the following ones: graph coloring and the Hamiltonian circuit problem. Chapter 1 presents basic definitions of graph theory, such as graph coloring, graph coloring with color-classes of bounded size b, and Hamiltonian circuits and paths. We also present online algorithms and online coloring. Chapter 2 starts with some general remarks about online graph covering with sets of bounded sizes (such as online bounded coloring): we give a simple method for transforming an online covering algorithm into an online bounded covering algorithm, and to derive the performance ratio of the bounded algorithm from the performance ratio of the unbounded algorithm. As will be shown in later chapters, this method often leads to optimal results. Furthermore, some basic preliminary results on online graph covering with sets of bounded size are given: for every graph, the performance ratio is bounded above by 1/2 + b/2 and for b = 2, this bound is optimal. In the second part, online coloring of co-interval graphs is studied. Based on two industrial applications, two different versions of this problem are discussed. In the case where the intervals are presented in increasing order of their left ends, we show that the performance ratio is 1 in the unbounded case and 2 - 1/b in the bounded case. In the case where the intervals may be presented in any order, we show that the performance ratio is at most 3 in the bounded case. Chapter 3 deals with online coloring of permutation and comparability graphs. First, we give a tight analysis of the First-Fit algorithm on bipartite permutation graphs and we show that its performance ratio is O(√n), even for some simple presentation orders. For both classes of graphs, we show that the performance ratio is bounded above by (χ+1)/2 in the unbounded case and that the performance ratio of First-Fit is equal to 1/2 + b/2 in the bounded case. In the second part of this chapter, we study cocoloring of permutation graphs. We show that the performance ratio is n/4 + 1/2 and we give better bounds in some more restricted cocoloring problems. Chapter 4 deals with an application of online coloring: the online Track Assignment Problem. Depending on the assumptions that are made, the Track Assignment Problem can be reduced to coloring permutation or overlap graphs online. We show that when a permutation graph is presented on a latticial plane, from west to east, then the performance ratio is exactly 2 - (min{b,k})-1, where k is the best known upper bound on the bounded chromatic number. We also show that, when a permutation graph is presented on a latticial plan, starting from the origin and growing, simultaneously or not, towards west and east, then the performance ratio is exactly 2 - 1/χ. We also show that online coloring overlap graphs does not have a performance ratio bounded by a constant, even if the overlap graph is bipartite and presented in increasing order of the intervals left ends. In this special case, we show that First-Fit has a tight performance ratio of O(√n). We consider coloring overlap graphs online where the intervals have a bounded size between 1 and a given number M. In this case, we show that the performance ratio can be bounded above by 2√M if M ≤ M0, and by log M (⎡log M / log log M⎤ + 1) if M > M0, M0 being defined by the equation 2√M0 = 3 log(M0). For large values of M, the ratio is O(log2 M / log log M). Chapter 5 is about online coloring of trees, forests and split-graphs. For trees, we show that the performance ratio of online coloring is exactly ½log2(2n) in the unbouded case and at most 1 + ⎣log2(b)⎦/χb in the bounded case. For split-graphs, we show that the performance ratio of online coloring is exactly 1 + 1/χ in the unbounded case and is at most 2 + 1/χb + 3/b in the bounded case. In Chapter 6, we present a class of digraphs: the quasi-adjoint graphs. These are a super class of both the graphs used for a DNA sequencing algorithm in (Blazewicz, Kasprzak, Computational complexity of isothermic DNA sequencing by hybridization, 2006) and the adjoints. A polynomial recognition algorithm in O(n3), as well as a polynomial algorithm in O(n2 + m2) for finding a Hamiltonian circuit in quasi-adjoint graphs are given. Furthermore, some results about related problems such as finding a Eulerian circuit while respecting some forbidden transitions (a sequence of two consecutive arcs) are discussed.

Distance spectrum, 1-factor and vertex-disjoint cycles

Vertex-disjoint cycles in bipartite tournaments

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian completion problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

Universal elements and the complexity of certain classes of infinite graphs

An extension of a theorem on cycles containing specified independent edges

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.