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DISSECTIONS OF POLYGONS INTO TRIANGLES OF EQUAL AREAS
Su, Zhanjun,Ding, Ren 한국전산응용수학회 2003 Journal of applied mathematics & informatics Vol.13 No.1
In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1990 it was proved that the statement is true for any centrally symmetric polygon. In the present paper we consider dissections of general polygons into triangles of equal areas.
LOCAL AND MEAN k-RAMSEY NUMBERS FOR THE FAMILY OF GRAPHS
Su, Zhanjun,Chen, Hongjing,Ding, Ren The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.3
For a family of graphs $\mathcal{H}$ and an integer k, we denote by $R^k(\mathcal{H})$ the corresponding k-Ramsey number, which is defined to be the smallest integer n such that every k-coloring of the edges of $K_n$ contains a monochromatic copy of a graph in $\mathcal{H}$. The local k-Ramsey number $R^k_{loc}(\mathcal{H})$ and the mean k-Ramsey number $R^k_{mean}(\mathcal{H})$ are defined analogously. Let $\mathcal{G}$ be the family of non-bipartite graphs and $T_n$ be the family of all trees on n vertices. In this paper we prove that $R^k_{loc}(\mathcal{G})=R^k_{mean}(\mathcal{G})$, and $R^2(T_n)$ < $R^2_{loc}(T_n)4 = $R^2_{mean}(T_n)$ for all $n\;{\ge}\;3$.
TILINGS OF PARALLELOGRAMS WITH SIMILAR TRIANGLES
Su, Zhanjun,Ding, Ren 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
We say that a triangle ${\Delta}$ tiles the polygon ${\rho}\;if\;{\rho}$ can be decomposed into finitely many non-overlapping triangles similar to ${\Delta}$. Let ${\rho}$ be a parallelogram with angles ${\delta}\;and\;{\pi}-{\delta}\;(0<{\delta}{\leq}{\pi}/2)$ and let ${\Delta}$ be a triangle with angles ${\alpha};{\beta},\;{\gamma}\;({\alpha}{\leq}{\beta}{\leq}{\gamma})$. We prove that if ${\Delta}$ tiles ${\rho}$ then either ${\delta}{\in}\;({\alpha},\;{\beta},\;{\gamma},\;{\pi}-{\gamma},\;{\pi}-2{\gamma})\;or\;dimL_{\rho}=dimL_{{\Delta}}$. We also prove that for every parallelogram P, and for every integer n $(where\;n{\geq}2,\;n{\neq}3)$ there is a triangle ${\Delta}$ so that n similar copies of ${\Delta}\;tile\;{\rho}$.
TILINGS OF ORTHOGONAL POLYGONS WITH SIMILAR RECTANGLES OR TRIANGLES
SU, ZHANJUN,DING, REN 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1
In this paper we prove two results about tilings of orthogonal polygons. (1) P be an orthogonal polygon with rational vertex coordinates and let R(u) be a rectangle with side lengths u and 1. An orthogonal polygon P can be tiled with similar copies of R(u) if and only if u i algebraic and the real part of each of its conjugates is positive; (2) Laczkovich proved that if a triangle $\Delta$ tiles a rectangle then either $\Delta$ is a right triangle or the angles of $\Delta$ are rational multiples of $\pi$. We generalize the result of Laczkovich to orthogonal polygons.
Tilings of orthogonal polygons with similarrectangles or triangles
Zhanjun Su,Ren Ding 한국전산응용수학회 2005 Journal of applied mathematics & informatics Vol.17 No.1-2
In this paper we prove two results about tilings of orthogonal polygons. (1) Let P be an orthogonal polygon with rational vertex coordinates and let R(u) be a rectangle with side lengths u and 1. An orthogonal polygon P can be tiled with similar copies of R(u) if and only if u is algebraic and the real part of each of its conjugates is positive; (2) Laczkovich proved that if a triangle tiles a rectangle then either is a right triangle or the angles of are rational multiples of . We generalize the result of Laczkovich to orthogonal polygons.
Tilings of parallelograms with similar triangles
ZHANJUN SU,REN DING 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
We say that a triangle tiles the polygon P if P can be decomposed into finitely many non-overlapping triangles similar to . Let P be a parallelogram with angles and − (0 < /2) and let be a triangle with angles , , ( ). We prove that if tiles P then either 2 {, , , − , −2 } or dimLP =dimL. We also prove that for every parallelogram P, and for every integer n (where n 2, n 6= 3) there is a triangle so that n similar copies of tile P.
LOCAL AND MEAN k-RAMSEY NUMBERS FOR THE FAMILY OF GRAPHS
Zhanjun Su,Hongjing Chen,Ren Ding 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.3
For a family of graphs H and an integer k, we denote by Rk(H) the corresponding k-Ramsey number, which is defined to be the smallest integer n such that every k-coloring of the edges of Kn contains a monochromatic copy of a graph in H. The local k- Ramsey number Rk loc(H) and the mean k-Ramsey number Rk mean(H) are defined analogously. Let G be the family of non-bipartite graphs and Tn be the family of all trees on n vertices. In this paper we prove that Rk loc(G) =Rk mean(G), and R2(T n) < R2 loc(T n) = R2 mean(T n) for all ≥ 3. For a family of graphs H and an integer k, we denote by Rk(H) the corresponding k-Ramsey number, which is defined to be the smallest integer n such that every k-coloring of the edges of Kn contains a monochromatic copy of a graph in H. The local k- Ramsey number Rk loc(H) and the mean k-Ramsey number Rk mean(H) are defined analogously. Let G be the family of non-bipartite graphs and Tn be the family of all trees on n vertices. In this paper we prove that Rk loc(G) =Rk mean(G), and R2(T n) < R2 loc(T n) = R2 mean(T n) for all ≥ 3.