148 research outputs found

    Hamilton cycles in 5-connected line graphs

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    A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness

    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

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Proceedings of the 10th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications

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    Proceedings of the 17th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

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    Winthrop University Undergraduate Scholarship & Creative Activity 2018

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    University College and Winthrop University proudly present Undergraduate Scholarship and Creative Activity 2018. This seventh annual University-wide compilation of undergraduate work chronicles the accomplishments of students and faculty mentors from at least 32 academic departments and programs, spanning all five colleges of the university: College of Arts and Sciences (CAS), College of Business Administration (CBA), College of Education (COE), College of Visual and Performing Arts (CVPA) and University College (UC).https://digitalcommons.winthrop.edu/undergradresearch_abstractbooks/1016/thumbnail.jp

    Seventh Biennial Report : June 2003 - March 2005

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