Long properly colored cycles in edge colored complete graphs

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A properly colored cycle (path) in $K_{n}^{c}$ 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 $\Delta^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if $\Delta^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored cycle of length at least $\lceil \frac{n+2}{3}\rceil+1$. In this paper, we improve the bound to $\lceil \frac{n}{2}\rceil + 2$.Comment: 8 page

A look at cycles containing specified elements of a graph

AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration

Kernels for Deletion to Classes of Acyclic Digraphs

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D - S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k^3) to O(k^2) and of Pumpkin Vertex Deletion Set from O(k^18) to O(k^3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions

Search-Space Reduction via Essential Vertices

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution

Towards a Polynomial Kernel for Directed Feedback Vertex Set

In the DIRECTED FEEDBACK VERTEX SET (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus

Extremal graph colouring and tiling problems

In this thesis, we study a variety of different extremal graph colouring and tiling problems in finite and infinite graphs. Confirming a conjecture of Gyárfás, we show that for all k, r ∈ N there is a constant C > 0 such that the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a collection of at most C monochromatic tight cycles. We shall say that the family of tight cycles has finite r-colour tiling number. We further prove that, for all natural numbers k, p and r, the family of p-th powers of k-uniform tight cycles has finite r-colour tiling number. The case where k = 2 settles a problem of Elekes, Soukup, Soukup and Szentmiklóssy. We then show that for all natural numbers ∆, r, every family F = {F1, F2, . . .} of graphs with v (Fn) = n and ∆(Fn) ≤ ∆ for every n ∈ N has finite r-colour tiling number. This makes progress on a conjecture of Grinshpun and Sárközy. We study Ramsey problems for infinite graphs and prove that in every 2-edge- colouring of KN, the countably infinite complete graph, there exists a monochromatic infinite path P such that V (P) has upper density at least (12 + √8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdős and Galvin. We study similar problems for many other graphs including trees and graphs of bounded degree or degeneracy and prove analogues of many results concerning graphs with linear Ramsey number in finite Ramsey theory. We also study a different sort of tiling problem which combines classical problems from extremal and probabilistic graph theory, the Corrádi–Hajnal theorem and (a special case of) the Johansson–Kahn–Vu theorem. We prove that there is some constant C > 0 such that the following is true for every n ∈ 3N and every p ≥ Cn−2/3 (log n)1/3. If G is a graph on n vertices with minimum degree at least 2n/3, then Gp (the random subgraph of G obtained by keeping every edge independently with probability p) contains a triangle tiling with high probability

On approximability and LP formulations for multicut and feedback set problems

Graph cut algorithms are an important tool for solving optimization problems in a variety of areas in computer science. Of particular importance is the min $s$-$t$ cut problem and an efficient (polynomial time) algorithm for it. Unfortunately, efficient algorithms are not known for several other cut problems. Furthermore, the theory of NP-completeness rules out the existence of efficient algorithms for these problems if the $P\neq NP$ conjecture is true. For this reason, much of the focus has shifted to the design of approximation algorithms. Over the past 30 years significant progress has been made in understanding the approximability of various graph cut problems. In this thesis we further advance our understanding by closing some of the gaps in the known approximability results. Our results comprise of new approximation algorithms as well as new hardness of approximation bounds. For both of these, new linear programming (LP) formulations based on a labeling viewpoint play a crucial role. One of the problems we consider is a generalization of the min $s$-$t$ cut problem, known as the multicut problem. In a multicut instance, we are given an undirected or directed weighted supply graph and a set of pairs of vertices which can be encoded as a demand graph. The goal is to remove a minimum weight set of edges from the supply graph such that all the demand pairs are disconnected. We study the effect of the structure of the demand graph on the approximability of multicut. We prove several algorithmic and hardness results which unify previous results and also yield new results. Our algorithmic result generalizes the constant factor approximations known for the undirected and directed multiway cut problems to a much larger class of demand graphs. Our hardness result proves the optimality of the hitting-set LP for directed graphs. In addition to the results on multicut, we also prove results for multiway cut and another special case of multicut, called linear-3-cut. Our results exhibit tight approximability bounds in some cases and improve upon the existing bound in other cases. As a consequence, we also obtain tight approximation results for related problems. Another part of the thesis is focused on feedback set problems. In a subset feedback edge or vertex set instance, we are given an undirected edge or vertex weighted graph, and a set of terminals. The goal is to find a minimum weight set of edges or vertices which hit all of the cycles that contain some terminal vertex. There is a natural hitting-set LP which has an $\Omega(\log k)$ integrality gap for $k$ terminals. Constant factor approximation algorithms have been developed using combinatorial techniques. However, the factors are not tight, and the algorithms are sometimes complicated. Since most of the related problems admit optimal approximation algorithms using LP relaxations, lack of good LP relaxations was seen as a fundamental roadblock towards resolving the approximability of these problems. In this thesis we address this by developing new LP relaxations with constant integrality gaps for subset feedback edge and vertex set problems