By Karim Adiprasito, Imre Bárány, Costin Vilcu

This quantity offers easy-to-understand but magnificent homes received utilizing topological, geometric and graph theoretic instruments within the parts lined through the Geometry convention that happened in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu at the party of his seventieth anniversary. The contributions tackle matters in convexity and discrete geometry, in distance geometry or with geometrical taste in combinatorics, graph concept or non-linear research. Written via most sensible specialists, those papers spotlight the shut connections among those fields, in addition to ties to different domain names of geometry and their reciprocal impression. they provide an summary on contemporary advancements in geometry and its border with discrete arithmetic, and supply solutions to numerous open questions. the quantity addresses a wide viewers in arithmetic, together with researchers and graduate scholars drawn to geometry and geometrical problems.

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**Sample text**

Yuan Fig. 5 u ∈ (1, 2] For the sake of convenience, we call an acute triangulation described in the proof of Case 1 (resp. Case 2) a type I (resp. type II ) acute triangulation of R. For any real number x, let [x] denote the greatest integer less than or equal to x and let {x} = x − [x]. 2 Any rectangle R with u > 2 admits an acute triangulation T with δT ≈ π4 and ⎧ 4u, ⎪ ⎪ ⎪ ⎨ 4u + 4, |T | = 8 u + 1 , ⎪ 2 ⎪ ⎪ ⎩ 16 u2 + 1 , where u 0 = i f u is even; i f u is odd; √ i f {u} > 0 and u 0 ∈ [ 2, 2); √ i f {u} > 0 and u 0 ∈ (1, 2), u .

Gardner, Mathematical games. The games and puzzles of Lewis Carroll, and the answers to February’s problems. Sci. Am. 202(3), 172–182 (1960) 6. M. C, Mathematical Association of America, 1995) 7. J. Itoh, L. Yuan, Acute triangulations of flat tori. Eur. J. Comb. 30, 1–4 (2009) 8. T. Hangan, J. Itoh, T. Zamfirescu, Acute triangulations. Bull. Math. Soc. Sci. Math. Roumanie 43(3–4), 279–285 (2000) 9. J. Itoh, T. Zamfirescu, Acute triangulations of the regular dodecahedral surface. Eur. J. Comb. 28, 1072–1086 (2007) 10.

He also established that every graph is the intersection graph of some family of subsets of a finite set. A lot of research has been done on various concepts that represent certain types of intersection graphs. Among these is the interval graph, (F), where U = , the real line, and each set Fi in F is an interval; certain interval graphs with various sorts of restrictions, such as unit-interval graphs, and multiple interval graphs; n-dimensional interval graph; circular-arc graph, etc. The monograph written by Mc Kee and Mc Morris [31] on Intersection Graph Theory is an excellent resource, as well as a good reference for most notations used in this paper.