Extremal Colorings and Independent Sets

We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. It was conjectured that extremal graphs are those which have clique number k and size (k2)+n−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(k2)+n−k(k2)+n−k. We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is Kk∪En−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eKk∪En−kKk∪En−k, K1∨(Kk−1∪En−k) role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eK1∨(Kk−1∪En−k)K1∨(Kk−1∪En−k) and (K1∨(Kk−1∪En−k−c+1))∪Ec−1 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(K1∨(Kk−1∪En−k−c+1))∪Ec−1(K1∨(Kk−1∪En−k−c+1))∪Ec−1 respectively

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

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

Sufficient degree conditions for graph embeddings

In this dissertation, we focus on the sufficient conditions to guarantee one graph being the subgraph of another. In Chapter 2, we discuss list packing, a modification of the idea of graph packing. This is fitting one graph in the complement of another graph. Sauer and Spencer showed a sufficient bound involving maximum degrees, and this was further explored by Kaul and Kostochka to characterize all extremal cases. Bollobas and Eldridge (and independently Sauer and Spencer) developed edge sum bounds to guarantee packing. In Chapter 2, we introduce the new idea of list packing and use it to prove stronger versions of many existing theorems. Namely, for two graphs, if the product of the maximum degrees is small or if the total number of edges is small, then the graphs pack. In Chapter 3, we discuss the problem of finding k vertex-disjoint cycles in a multigraph. This problem originated from a conjecture of Erdos and has led to many different results. Corradi and Hajnal looked at a minimum degree condition. Enomoto and Wang independently looked at a minimum degree-sum condition. More recently, Kierstead, Kostochka, and Yeager characterized the extremal cases to improve these bounds. In Chapter 3, we improve on the multigraph degree-sum result. We characterize all multigraphs that have simple Ore-degree at least 4k -3 , but do not contain k vertex-disjoint cycles. Moreover, we provide a polynomial time algorithm for deciding if a graph contains k vertex-disjoint cycles. Lastly, in Chapter 4, we consider the same problem but with chorded cycles. Finkel looked at the minimum degree condition while Chiba, Fujita, Gao, and Li addressed the degree-sum condition. More recently, Molla, Santana, and Yeager improved this degree-sum result, and in Chapter 4, we will improve on this further

On the integrity of domination in graphs.

Thesis (M.Sc.)-University of Natal, 1993.This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to which domination properties of a graph are preserved if the graph is altered by the deletion of vertices or edges or by the insertion of new edges. A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge-( domination-)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E E(G). We explore fundamental properties of such graphs and their characterization for small values of k. Particular attention is devoted to 3-edge-critical graphs. In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated. \ Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that )'(G-5) > )'(G) ()'(G -5) < )'(G), respectively), are investigated, as are'k-vertex-critical graphs G (with )'(G) = k and )'(G-v) < k for all v E V(O)). The existence of smallest'domination-forcing sets of vertices of graphs is considered. The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that )'(G- F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are considered, with particular reference to such graphs of minimum size. Finally, in Chapter 5, we-discuss n-dominating setsD of a graph G (such that each vertex in G-D is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and fourth contain a listing of unsolved problems and conjectures

Quantum nonlocality, cryptography and complexity

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