Conditions for ß-perfectness

A ß-perfect graph is a simple graph G such that ¿(G') = ß(G') for every induced subgraph G' of G, where ¿(G') is the chromatic number of G', and ß(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., d(H)+1). The vertices of a ß-perfect graph G can be coloured with ¿(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs, for a graph to be ß-perfect. We give new sufficient conditions and make improvements to sufficient conditions previously given by others. We also mention a necessary condition which generalizes the fact that no ß-perfect graph contains an even hole

β-Perfect Graphs

AbstractThe class ofβ-perfect graphs is introduced. We draw a number of parallels between these graphs and perfect graphs. We also introduce some special classes ofβ-perfect graphs. Finally, we show that the greedy algorithm can be used to colour a graphGwith no even chordless cycles using at most 2(χ(G)−1) colours

On vertex neighborhood in minimal imperfect graphs

AbstractLubiw (J. Combin. Theory Ser. B 51 (1991) 24) conjectures that in a minimal imperfect Berge graph, the neighborhood graph N(v) of any vertex v must be connected; this conjecture implies a well known Chvátal's conjecture (Chvátal, First Workshop on Perfect Graphs, Princeton, 1993) which states that N(v) must contain a P4. In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P4-free if, and only if, every induced subgraph with at least two vertices either is disconnected or its complement is disconnected; this characterization implies that P4-free graphs can be constructed by complete join and disjoint union from isolated vertices. We propose to replace P4-free graphs by a similar construction using bipartite graphs instead of isolated vertices

Capacity and coding in digital communications

+164hlm.;24c

Discrepancy Inequalities in Graphs and Their Applications

Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based not on the spectral gap of a graph, but rather a different tool with motivations in Riemannian geometry. The first problem explored in this dissertation is motivated by parallel computing and other communication networks. Consider a connected graph G, with a pebble placed on each vertex of G. The routing number, rt(G), of G is the minimum number of steps needed to route any permutation on the vertices of G, where a step consists of selecting a matching in the graph and swapping the pebbles on the endpoints of each edge. Alon, Chung, and Graham introduced this parameter, and (among other results) gave a bound based on the spectral gap for general graphs. The bound they obtain is polylogarithmic for graphs with a sufficiently strong spectral gap. In this dissertation, we use the Expander Mixing Lemma, the probablistic method, and other extremal tools to investigate when this upper bound can be improved to be constant depending on the gap and the vertex degrees. The second problem examined in this dissertation has motivations in a question of Erdõs and Pósa, who conjectured that every sufficiently dense graph on n vertices, where n is divisible by 3, decomposes into triangles. While Corradi and Hajnal proved this result true for graphs with minimum degree at least (2/3)n, their result spawned a series of similar questions about the number of vertex-disjoint subgraphs of a certain class that a graph with some degree condition must contain. While this problem is well-studied for dense graphs, many results give significantly worse bounds for less dense graphs. Using spectral graph theory, we show that every graph with some weak density and spectral conditions contains O(sqrt(nd)) vertex-disjoint cycles. Furthermore, even if we require these cycles to contain a certain number of chords, a graph satisfying these conditions will still contain O(sqrt(nd)) such vertex-disjoint cycles. In both cases, we show this bound to be best possible. Finally, we conclude by obtaining local version of a discrepancy inequality. An oversimplification of the Expander Mixing Lemma states that a graph with a strong spectral condition must have nice edge distribution. We seek to mimic that idea, but by using discrete curvature instead of a spectral condition. Discrete curvature, inspired by its counterpart in Riemannian geometry, measures the local volume growth at a vertex. Thus, given a vertex x, our result uses curvature to quantify the edge distribution between vertices that are a distance one from x and vertices that are a distance two from x. In doing this, we are able to study the number of 3-cycles and 4-cycles containing a particular edge