29 research outputs found

    Local properties of graphs with large chromatic number

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    This thesis deals with problems concerning the local properties of graphs with large chromatic number in hereditary classes of graphs. We construct intersection graphs of axis-aligned boxes and of lines in R3\mathbb{R}^3 that have arbitrarily large girth and chromatic number. We also prove that the maximum chromatic number of a circle graph with clique number at most ω\omega is equal to Θ(ωlogω)\Theta(\omega \log \omega). Lastly, extending the χ\chi-boundedness of circle graphs, we prove a conjecture of Geelen that every proper vertex-minor-closed class of graphs is χ\chi-bounded

    2016-2017, University of Memphis bulletin

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    University of Memphis bulletin containing the graduate catalog for 2016-2017.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1436/thumbnail.jp

    2015-2016, University of Memphis bulletin

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    University of Memphis bulletin containing the graduate catalog for 2015-2016.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1435/thumbnail.jp

    A survey of χ\chi-boundedness

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    If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now

    Chi-boundedness of graph classes excluding wheel vertex-minors

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    A class of graphs is χ-bounded if there exists a function f:N→N such that for every graph G in the class and an induced subgraph H of G, if H has no clique of size q+1, then the chromatic number of H is less than or equal to f(q). We denote by Wn the wheel graph on n+1 vertices. We show that the class of graphs having no vertex-minor isomorphic to Wn is χ-bounded. This generalizes several previous results; χ-boundedness for circle graphs, for graphs having no W5 vertex-minors, and for graphs having no fan vertex-minors

    Cliques, Degrees, and Coloring: Expanding the ω, Δ, χ paradigm

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    Many of the most celebrated and influential results in graph coloring, such as Brooks' Theorem and Vizing's Theorem, relate a graph's chromatic number to its clique number or maximum degree. Currently, several of the most important and enticing open problems in coloring, such as Reed's ω,Δ,χ\omega, \Delta, \chi Conjecture, follow this theme. This thesis both broadens and deepens this classical paradigm. In Part~1, we tackle list-coloring problems in which the number of colors available to each vertex vv depends on its degree, denoted d(v)d(v), and the size of the largest clique containing it, denoted ω(v)\omega(v). We make extensive use of the probabilistic method in this part. We conjecture the ``list-local version'' of Reed's Conjecture, that is every graph is LL-colorable if LL is a list-assignment such that L(v)(1ε)(d(v)+1)+εω(v))|L(v)| \geq \lceil (1 - \varepsilon)(d(v) + 1) + \varepsilon\omega(v))\rceil for each vertex vv and ε1/2\varepsilon \leq 1/2, and we prove this for ε1/330\varepsilon \leq 1/330 under some mild additional assumptions. We also conjecture the ``mad\mathrm{mad} version'' of Reed's Conjecture, even for list-coloring. That is, for ε1/2\varepsilon \leq 1/2, every graph GG satisfies \chi_\ell(G) \leq \lceil (1 - \varepsilon)(\mad(G) + 1) + \varepsilon\omega(G)\rceil, where mad(G)\mathrm{mad}(G) is the maximum average degree of GG. We prove this conjecture for small values of ε\varepsilon, assuming ω(G)mad(G)log10mad(G)\omega(G) \leq \mathrm{mad}(G) - \log^{10}\mathrm{mad}(G). We actually prove a stronger result that improves bounds on the density of critical graphs without large cliques, a long-standing problem, answering a question of Kostochka and Yancey. In the proof, we use a novel application of the discharging method to find a set of vertices for which any precoloring can be extended to the remainder of the graph using the probabilistic method. Our result also makes progress towards Hadwiger's Conjecture: we improve the best known bound on the chromatic number of KtK_t-minor free graphs by a constant factor. We provide a unified treatment of coloring graphs with small clique number. We prove that for Δ\Delta sufficiently large, if GG is a graph of maximum degree at most Δ\Delta with list-assignment LL such that for each vertex vV(G)v\in V(G), L(v)72d(v)min{ln(ω(v))ln(d(v)),ω(v)ln(ln(d(v)))ln(d(v)),log2(χ(G[N(v)])+1)ln(d(v))}|L(v)| \geq 72\cdot d(v)\min\left\{\sqrt{\frac{\ln(\omega(v))}{\ln(d(v))}}, \frac{\omega(v)\ln(\ln(d(v)))}{\ln(d(v))}, \frac{\log_2(\chi(G[N(v)]) + 1)}{\ln(d(v))}\right\} and d(v)ln2Δd(v) \geq \ln^2\Delta, then GG is LL-colorable. This result simultaneously implies three famous results of Johansson from the 90s, as well as the following new bound on the chromatic number of any graph GG with ω(G)ω\omega(G)\leq \omega and Δ(G)Δ\Delta(G)\leq \Delta for Δ\Delta sufficiently large: χ(G)72ΔlnωlnΔ.\chi(G) \leq 72\Delta\sqrt{\frac{\ln\omega}{\ln\Delta}}. In Part~2, we introduce and develop the theory of fractional coloring with local demands. A fractional coloring of a graph is an assignment of measurable subsets of the [0,1][0, 1]-interval to each vertex such that adjacent vertices receive disjoint sets, and we think of vertices ``demanding'' to receive a set of color of comparatively large measure. We prove and conjecture ``local demands versions'' of various well-known coloring results in the ω,Δ,χ\omega, \Delta, \chi paradigm, including Vizing's Theorem and Molloy's recent breakthrough bound on the chromatic number of triangle-free graphs. The highlight of this part is the ``local demands version'' of Brooks' Theorem. Namely, we prove that if GG is a graph and f:V(G)[0,1]f : V(G) \rightarrow [0, 1] such that every clique KK in GG satisfies vKf(v)1\sum_{v\in K}f(v) \leq 1 and every vertex vV(G)v\in V(G) demands f(v)1/(d(v)+1/2)f(v) \leq 1/(d(v) + 1/2), then GG has a fractional coloring ϕ\phi in which the measure of ϕ(v)\phi(v) for each vertex vV(G)v\in V(G) is at least f(v)f(v). This result generalizes the Caro-Wei Theorem and improves its bound on the independence number, and it is tight for the 5-cycle

    Graph Theory

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    Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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