Study of Graph Partitions Approach satisfies Vizing's Conjecture

For a graph G(V,E) , a subset of vertices D is a dominating set if for each vertex 12x∈"> V either  12x∈D">  , or x is adjacent to at least one vertex of D . The domination number , 12خ³G, "> is the order of smallest dominating set of G . In [7],  Vizing conjectured that 12خ³Gأ—H ≥ خ³Gأ—خ³H">  for any two graphs G and H , where G×H denotes their Cartesian product . This conjecture is still open . In this paper , we investigate following relations, if a graph H has a D-partition then it also has a K-partition, and if  H  has a K-partition , then Vizing's conjecture is satisfied for any graph G , after that, every cycle 12Cn , n≥3">  , has a K-partition. Moreover, if H has    a K-partition , then  H satisfies the following relations  12خ³H≤2"> , 12P2H=خ³H">  and H is a perfect-dominated graph .

Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

Edge-colourings of graphs

All the results in this thesis are concerned with the classification of graphs by their chromatic class. We first extend earlier results of Fiorini and others to give a complete list of critical graphs of order at most ten. We give conditions for extending the edge-colouring of a nearly complete subgraph to the whole graph and use this result to prove a special case of Vizing's conjecture. We also use other methods to solve further cases of this conjecture. A major part of the thesis classifies those graphs with at most 4 vertices of maximum degree and this work is generalised to graphs with r vertices of maximum degree. We also completely classify all regular graphs G with degree at least 6/7|V(G)|. Finally we give some examples of even order critical graphs and introduce the concept of a supersnark

The chromatic index of simple graphs

The object of this thesis is twofold: (i) to study the structural properties of graphs which are critical with respect to edge-colourings; (ii) to apply the results obtained to the classification problem arising from Vizing's Theorem. Chapter 1 contains a historical, non-technical introduction, general graph-theoretic definitions and notation, a discussion of Vizing's Theorem as well as a survey of the main results obtained to date in Vizing's classification problem. Chapter 2 introduces the notion of criticality in the first section; the second section contains both well-known and new constructions of critical graphs which will be used in later chapters. The third and final section contains new results concerning elementary properties of critical graphs. Chapter 3 deals with uniquely-colourable graphs and their relationship to critical graphs. Chapter 4 contains results on the connectivity of critical graphs, whereas Chapter 5 deals with bounds on the number of edges of these graphs. In particular, bounds improving those given by Vizing are presented. These results are applied to problems concerning planar graphs. In Chapter 6, critical graphs of small order are discussed. All such graphs of order at most 8 are determined, while the 'critical graph conjecture’ of Beineke & Wilson and Jakobsen is shown to be true for all graphs on at most 10 vertices. The seventh and final chapter deals with circuit length properties of critical graphs. In particular, the minimal order of certain critical graphs with given girth and maximum valency is determined. Results improving Vizing’s estimate of the circumference of critical graphs are also given. The Appendix includes a computer programme which generates critical graphs from simpler ones using a constructive algorithm given in Chapter 2

Kőnig's Line Coloring and Vizing's Theorems for Graphings

The classical theorem of Vizing states that every graph of maximum degree d admits an edge coloring with at most d+1 colors. Furthermore, as it was earlier shown by Kőnig, d colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree d admits a measurable edge coloring with d+O(d−−√) colors; furthermore, if the graphing has no odd cycles, then d+1 colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that d+1 colors are always enough

Extremal problems on counting combinatorial structures

The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on $n$ vertices has \floor{n^2/4} edges. Since all of its subgraphs are triangle-free, the number of (labeled) triangle-free graphs on $n$ vertices is at least 2^{\floor{n^2/4}}. This was shown to be the correct order of magnitude in a celebrated paper Erd\H{o}s, Kleitman, and Rothschild from 1976, where the authors furthermore proved that almost all triangle-free graphs are bipartite. In Chapters 2 and 3 we study analogous problems for triangle-free graphs that are maximal with respect to inclusion. In Chapter 2, we solve the following problem of Paul Erd\H{o}s: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. We show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among other tools the Ruzsa-Szemer\'{e}di Triangle Removal Lemma and recent results on characterizing of the structure of independent sets in hypergraphs. This is a joint work with J\'{o}zsef Balogh. In Chapter 3, we investigate the structure of maximal triangle-free graphs. We prove that almost all maximal triangle-free graphs admit a vertex partition $(X, Y)$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di Removal Lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on the characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest. This is a joint work with J\'{o}zsef Balogh, Hong Liu, and Maryam Sharifzadeh. In Chapte 4, we seek families in posets with the smallest number of comparable pairs. Given a poset $P$, a family \F\subseteq P is \emph{centered} if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the \emph{centeredness property} if for any $M$, among all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbb{F}_q^n$ has the centeredness property. Several open problems are also given. This is a joint result with J\'{o}zsef Balogh and Adam Zsolt Wagner. In Chapter 5, we consider a graph coloring problem. Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$-th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $\chi(G^k)$ unbounded and $\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k)$. We also provide an upper bound, $\chi_{\ell}(G^k)1$. This is a joint work with Nicholas Kosar, Benjamin Reiniger, and Elyse Yeager

Graph Structure and Coloring

We denote by G=(V,E) a graph with vertex set V and edge set E. A graph G is claw-free if no vertex of G has three pairwise nonadjacent neighbours. Claw-free graphs are a natural generalization of line graphs. This thesis answers several questions about claw-free graphs and line graphs. In 1988, Chvatal and Sbihi proved a decomposition theorem for claw-free perfect graphs. They showed that claw-free perfect graphs either have a clique-cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed successfully described how elementary graphs can be built using line graphs of bipartite graphs and local augmentation. However gluing two claw-free perfect graphs on a clique does not necessarily produce claw-free graphs. The first result of this thesis is a complete structural description of claw-free perfect graphs. We also give a construction for all perfect circular interval graphs. This is joint work with Chudnovsky. Erdos and Lovasz conjectured in 1968 that for every graph G and all integers s,t≥ 2 such that s+t-1=χ(G) > ω(G), there exists a partition (S,T) of the vertex set of G such that ω(G|S)≥ s and χ(G|T)≥ t. This conjecture is known in the graph theory community as the Erdos-Lovasz Tihany Conjecture. For general graphs, the only settled cases of the conjecture are when s and t are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2, line graphs and quasi-line graphs. The second part of this thesis considers the conjecture for claw-free graphs and presents some progresses on it. This is joint work with Chudnovsky and Fradkin. Reed's ω, ∆, χ conjecture proposes that every graph satisfies χ≤ ⎡½ (Δ+1+ω)⎤ ; it is known to hold for all claw-free graphs. The third part of this thesis considers a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colorings that achieve our bounds: The complexity are O(n²) for line graphs and O(n³m²) for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a coloring that achieves the bound of Reed's original conjecture. This is joint work with Chudnovsky, King and Seymour