Towards obtaining a 3-Decomposition from a perfect Matching

A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list. In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching $M$ from a cubic graph $G$ leaves a disjoint union of cycles. Contracting these cycles yields a new graph $G_M$. The graph $G$ is star-like if $G_M$ is a star for some perfect matching $M$, making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.Comment: 21 pages, 7 figure

Graph Partitioning With Input Restrictions

In this thesis we study the computational complexity of a number of graph partitioning problems under a variety of input restrictions. Predominantly, we research problems related to Colouring in the case where the input is limited to hereditary graph classes, graphs of bounded diameter or some combination of the two. In Chapter 2 we demonstrate the dramatic eect that restricting our input to hereditary graph classes can have on the complexity of a decision problem. To do this, we show extreme jumps in the complexity of three problems related to graph colouring between the class of all graphs and every other hereditary graph class. We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be H-free for some graph H if it contains no induced subgraph isomorphic to H. Similarly, G is said to be H-free for some set of graphs H, if it does not contain any graph in H as an induced subgraph. Here, the set H consists usually of a single cycle or tree but may also contain a number of cycles, for example we give results for graphs of bounded diameter and girth. Chapter 4 is dedicated to three variants of the Colouring problem, Acyclic Colouring, Star Colouring, and Injective Colouring. We give complete or almost complete dichotomies for each of these decision problems restricted to H-free graphs. In Chapter 5 we study these problems, along with three further variants of 3-Colouring, Independent Odd Cycle Transversal, Independent Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of bounded diameter. Finally, Chapter 6 deals with a dierent variety of problems. We study the problems Disjoint Paths and Disjoint Connected Subgraphs for H-free graphs

Homology groups for particles on one-connected graphs

We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces and some limited use of discrete Morse theory. As one of the results, we derive a closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles. Our motivation is the search for new kinds of quantum statistics.Comment: 26 pages, 16 figure

On globally sparse Ramsey graphs

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available

Restricted frame graphs and a conjecture of Scott

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our results to an appendix

Finding Cycles and Trees in Sublinear Time

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., $\Omega(1)$-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time \tildeO(\sqrt{N}), where $N$ denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of $N$-vertex graphs can be tested with one-sided error within time complexity \tildeO(\poly(1/\e)\cdot\sqrt{N}). This matches the known $\Omega(\sqrt{N})$ query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter \e. For any constant $k\geq3$, we extend this result to testing whether the input graph has a simple cycle of length at least $k$. On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter \e. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in $o(\sqrt{N})$ complexity.Comment: Keywords: Sublinear-Time Algorithms, Property Testing, Bounded-Degree Graphs, One-Sided vs Two-Sided Error Probability Updated versio

Computing Graph Roots Without Short Cycles

Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H2 for some graph H of small girth. The main results are the following. - There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism. - There is a polynomial time algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is NP-complete to recognize if G = H2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed