Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Bipartite Graph Packing Problems

The overarching problem of this project was trying to find the maximal number of disjoint subgraphs of a certain type we can pack into any graph. These disjoint graphs could be of any type in the original problem. However, they were limited to be T2 trees for my research (T2 trees are defined in section 2.1 of the paper). In addition, most of my work was focused on packing these T2 trees into constrained bipartite graphs (also defined in section 2.1 of the paper). Even with these specific constraints applied to the overall problem, the project still branched into different subproblems such as packing trees into complete bipartite graphs and finding minimal and maximal bounds for packing these graphs

Trees and graph packing

In this thesis we investigate two main topics, namely, suffix trees and graph packing problems. In Chapter 2, we present the suffix trees. The main result of this chapter is a lower bound on the size of simple suffix trees. In the rest of the thesis we deal with packing problems. In Chapter 3 we give almost tight conditions on a bipartite packing problem. In Chapter 4 we consider an embedding problem regarding degree sequences. In Chapter 5 we show the existence of bounded degree bipartite graphs with a small separator and large bandwidth and we prove that under certain conditions these graphs can be embedded into graphs with minimum degree slightly over n/2

Dynamic programming on bipartite tree decompositions

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that $K_t$-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by bipartite treewidth. We provide the following complexity dichotomy when $H$ is a 2-connected graph, for each of $H$-Subgraph-Packing, $H$-Induced-Packing, $H$-Scattered-Packing, and $H$-Odd-Minor-Packing problem: if $H$ is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if $H$ is non-bipartite, then it is solvable in XP-time. We define 1-${\cal H}$-treewidth by replacing the bipartite graph class by any class ${\cal H}$. Most of the technology developed here works for this more general parameter.Comment: Presented in IPEC 202

On the spanning tree packing number of a graph: a survey

AbstractThe spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes

A list version of graph packing

We consider the following generalization of graph packing. Let $G_{1} = (V_{1}, E_{1})$ and $G_{2} = (V_{2}, E_{2})$ be graphs of order $n$ and $G_{3} = (V_{1} \cup V_{2}, E_{3})$ a bipartite graph. A bijection $f$ from $V_{1}$ onto $V_{2}$ is a list packing of the triple $(G_{1}, G_{2}, G_{3})$ if $uv \in E_{2}$ implies $f(u)f(v) \notin E_{2}$ and $vf(v) \notin E_{3}$ for all $v \in V_{1}$. We extend the classical results of Sauer and Spencer and Bollob\'{a}s and Eldridge on packing of graphs with small sizes or maximum degrees to the setting of list packing. In particular, we extend the well-known Bollob\'{a}s--Eldridge Theorem, proving that if $\Delta (G_{1}) \leq n-2, \Delta(G_{2}) \leq n-2, \Delta(G_{3}) \leq n-1$, and $|E_1| + |E_2| + |E_3| \leq 2n-3$, then either $(G_{1}, G_{2}, G_{3})$ packs or is one of 7 possible exceptions. Hopefully, the concept of list packing will help to solve some problems on ordinary graph packing, as the concept of list coloring did for ordinary coloring.Comment: 10 pages, 4 figure

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rateɛ-balanced. Sharp bounds onɛ in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallestɛ in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,ɛ-balanced forɛ≈0.06

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved