Global ETD Search
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Showing 1 to 12 of 12 for “"Triangle-free"”.
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Chromatic Thresholds of Regular Graphs with Small Cliques
… results related to the chromatic threshold of triangle-free graphs have been reached in the last 13 years, culminating in a result by Brandt and Thomassé stating that any triangle-free graph on <em>n</em> vertices with minimum degree exceeding 1/3 <em>n</em> has chromatic number at most 4. In …
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Extremal problems on counting combinatorial structures
… 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 …
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Finding Edge and Vertex Induced Cycles within Circulants.
… to obtain an automorphic decomposition of a triangle-free graph. As many of their examples involve circulant graphs, it is of particular interest to find triangle-free subgraphs within circulants. As a cycle with at least four vertices is a canonical example of a triangle-free subgraph, we …
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Cliques in graphs
… fundamental result in Graph Theory states that a triangle-free graph of order $n$ has at most $n^2/4$ edges. Hence, a triangle-free graph has minimum degree at most $n/2$, so if $k_3(n,\delta) =0$ then $\delta \le n/2$. For $n/2 \leq \delta \leq 4n/5$, I have evaluated $k_r(n,\delta)$ and …
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Four Problems in Probability and Optimization
… also prove an integrality gap of $1/2$ for the triangle free problem, and show that at least $n/2$ steps are required for the triangle free problem's theta bodies to converge in the case $G = K_n$. We introduce a criterion for an invariant polynomial to be a sum of squares on the hypercube. This …
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Extremal problems involving forbidden subgraphs
… we find the exact threshold of density of triangle-free $(0,k)$-colorable graphs and we find the asymptotic threshold of density of $(j,k)$-colorable graphs of large girth when $k\geq 2j+2$. In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic …
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Fractional Chromatic Numbers and Spectra of Graphs
… Based on the study of independence numbers of triangle-free graphs with maximum degree at most three, Heckman and Thomas conjectured that $\chi_f(G) \leq 3-\frac{1}{5}$ if $G$ is triangle-free and has maximum degree at most three. Since the fractional chromatic number of the generalized …
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Enumerating combinatorial objects with limited sub-configurations
… problem on Gallai colorings, i.e. rainbow triangle-free colorings. In particular, we describe the typical structure of Gallai r-colorings of complete graphs, and complete the characterization of the extremal graphs for Gallai colorings. This work heavily relies on the hypergraph container …
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Games on Graphs and Other Combinatorial Problems
… Pach, Pollak and Tuza: given a connected, triangle-free graph on $n$ vertices and of minimum degree at least $\delta$, how large can the radius of such a graph be? We also study the variant of this problem in which the triangle-free condition is replaced by a condition about the girth of …
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The Chromatic Structure of Dense Graphs
… determining the minimum degree stability of H-free graphs, the focus of Chapter 5. Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and Simonovits. …
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Decay of correlations and inference in graphical models
… for this model, for arbitrary bounded degree triangle-free graphs. Our results hold for any [alpha] > [alpha]* whenever the size of the list of each vertex v is at least [alpha][delta](v) + [beta] where [delta](v) is the degree of vertex v and [beta] is a constant that only depends on [alpha]. …