Our main result is a rigidity theorem: we show that if sampling two measures induce the same probability distribution on order types, then these measures are projectively equivalent provided the support of at least one of them has non-empty interior. Next we establish results on the analytic representation of limits of order types by planar measures. Using flag algebras we obtain new numerical results on the Erdős problem of finding the minimal density of 5-or 6-tuples in convex position in an arbitrary point set, and also an inequality expressing the difficulty of sampling order types uniformly. We apply ideas from the theory of limits of dense combinatorial structures to study order types, which are combinatorial encodings of finite point sets. We do not use the discharging method but rather exploit decomposition trees of $K_4$-minor-free graphs.ģ1st International Symposium on Computational Geometry (SOCG 2015) arxiv:1811.02236 Abstract This bound is tight and confirms a conjecture by Zhang and Whu. We demonstrate that for every positive integer $\Delta$, every $K_4$-minor-free graph with maximum degree $\Delta$ admits an equitable coloring with k colors wherek $\ge$ ($\Delta$+3)/2. Journal of Graph Algorithms and Applications arxiv:1703.02250 Abstract Equitable Colorings of $K_4$-minor-free Graphs.JOHNSON AT& T Bell Laboratories, Murray Hill, NJ, USA Received 2 December 1988 Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in computational. COLBOURN Department of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 3Gl Canada David S. Graf Gra95 obtains a tighter bound on the chromatic number. Discrete Mathematics 86 (1990) 165-177 165 North-Holland UNIT DISK GRAPHS Brent N. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erdős and Nešetřil. NP-complete for the intersection graphs of unit disks in the plane (unit disk graphs), a. Is there some absolute $\varepsilon > 0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\chi(G^2) \le (2-\varepsilon) \omega(G)^2$, where $\omega(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case of $G$ the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom_p$-flow in which each arc is assigned a value of its own list.Ĭanadian Journal of Mathematics arxiv:1609.08645 Abstract An improved procedure for colouring graphs of bounded local density.International Conference on Machine Learning (ICML 2021) Improving Ultrametrics Embeddings Through Coresets.Property testing, computational geometry andĪpplications of limits of combinatorial structures in geometry and graph theory Scientific publications The thesis explores the connections of three topics with limits of combinatorial objects: Graph theory, and particularly graph coloring.In the past, I worked for instance on the following topics: Unbounded from Monopolarity assuming Polynomial,NP-complete disjoint.I am interested in all aspects of theoretical computer science and discrete mathematics. Its distance to clique is the minimum number of vertices that have to be deleted from $G$ in order to obtain a clique. Unbounded from Weighted independent set assuming Polynomial,NP-complete disjoint. Unbounded from Weighted independent dominating set assuming Polynomial,NP-complete disjoint. Unbounded from Weighted feedback vertex set assuming Polynomial,NP-complete disjoint. Unbounded from Polarity assuming Polynomial,NP-complete disjoint. Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint. the corresponding unit-disk graph is a co-comparability graph. Unbounded from Independent set assuming Polynomial,NP-complete disjoint. Recognition of graphs with no (k) independent cycles (2022) We say that a graph is. The best known lower bound for the chromatic number of the unit distance graph of Euclidean n-space is by Raigorodskii (Electronic Notes in Discrete. Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint. Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint. Unbounded from Feedback vertex set assuming Polynomial,NP-complete disjoint. Unbounded from Domination assuming Polynomial,NP-complete disjoint. Unbounded from Colourability assuming Polynomial,NP-complete disjoint. Unbounded from Clique cover assuming Polynomial,NP-complete disjoint. Unbounded from 3-Colourability assuming Polynomial,NP-complete disjoint. Is a bijection from $V(G)$ to the leaves of the tree $T$. Consider the following decomposition of a graph $G$ which is defined as a pair $(T,L)$ where $T$ is a binary tree and $L$
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