A near-triangular embedding is an embedded graph into some surface whose all but one facial walks are 3-gons. In this paper we show that if a graph G is a triangulation of an orientable surface Sh, then G has a near-t...A near-triangular embedding is an embedded graph into some surface whose all but one facial walks are 3-gons. In this paper we show that if a graph G is a triangulation of an orientable surface Sh, then G has a near-triangular embedding into Sk for k=h, h＋1,...1,[β（G）/2], where β（G） is the Betti number of G.展开更多
An orientation of a graph G with even number of vertices is Pfaffian if every even cycle C such that G-V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance...An orientation of a graph G with even number of vertices is Pfaffian if every even cycle C such that G-V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations stems from the fact that if a graph G has one, then the number of perfect matchings of G can be computed in polynomial time. There is a classical result of Kasteleyn that every planar graph has a Pfaffian orientation. Little proved an elegant characterization of bipartite graphs that admit a Pfaffian orientation. Robertson, Seymour and Thomas (1999) gave a polynomial-time recognition algorithm to test whether a bipartite graph is Pfaffian by a structural description of bipartite graphs. In this paper, we consider the Pfaffian property of graphs embedding on the orientable surface with genus one (i.e., the torus). Some sufficient conditions for Pfaffian graphs on the torus are obtained. Furthermore, we show that all quadrilateral tilings on the torus are Pfaffian if and only if they are not bipartite graphs.展开更多
The embedding technique based on an operator appeared in [Liu, Y. P., Scientia Sinica, Special Issue on Math,1 (1979),191-201 (in Chinese)] for determining the maximum non-orientable genus of a graph is developed to o...The embedding technique based on an operator appeared in [Liu, Y. P., Scientia Sinica, Special Issue on Math,1 (1979),191-201 (in Chinese)] for determining the maximum non-orientable genus of a graph is developed to obtain the general theorem which presents a necessary and sufficient condition for a graph to be embeddable into either the orientable or the non-orientable surface of genus k. Furthermore,the greatest lower bound of the lengths of genus ranges of the class of nonplanar graphs which are up-embeddable is also obtained.展开更多
Call a periodic map h on the closed orientable surface Σg extendable if h extends to a periodic map over the pair(S3, Σg) for possible embeddings e : Σg→ S3. The authors determine the extendabilities for all perio...Call a periodic map h on the closed orientable surface Σg extendable if h extends to a periodic map over the pair(S3, Σg) for possible embeddings e : Σg→ S3. The authors determine the extendabilities for all periodical maps on Σ2. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair(S3, Σg). To do this the authors first list all periodic maps on Σ2, and indeed the authors exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be interesting. A by-product is that for each even g,the maximum order periodic map on Σg is extendable, which contrasts sharply with the situation in the orientation preserving category.展开更多
A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric e...A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.展开更多
In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficie...In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficiently small.展开更多
基金the National Natural Science Foundation of China (19831080)Shanghai City Fundation of Selected Academic Research (04JC14031)
文摘A near-triangular embedding is an embedded graph into some surface whose all but one facial walks are 3-gons. In this paper we show that if a graph G is a triangulation of an orientable surface Sh, then G has a near-triangular embedding into Sk for k=h, h＋1,...1,[β（G）/2], where β（G） is the Betti number of G.
基金National Natural Science Foundation of China (Grant Nos. 10831001 and 11171279)the Scientific Research Foundation of Zhangzhou Normal University (Grant No. SX1002)
文摘An orientation of a graph G with even number of vertices is Pfaffian if every even cycle C such that G-V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations stems from the fact that if a graph G has one, then the number of perfect matchings of G can be computed in polynomial time. There is a classical result of Kasteleyn that every planar graph has a Pfaffian orientation. Little proved an elegant characterization of bipartite graphs that admit a Pfaffian orientation. Robertson, Seymour and Thomas (1999) gave a polynomial-time recognition algorithm to test whether a bipartite graph is Pfaffian by a structural description of bipartite graphs. In this paper, we consider the Pfaffian property of graphs embedding on the orientable surface with genus one (i.e., the torus). Some sufficient conditions for Pfaffian graphs on the torus are obtained. Furthermore, we show that all quadrilateral tilings on the torus are Pfaffian if and only if they are not bipartite graphs.
文摘The embedding technique based on an operator appeared in [Liu, Y. P., Scientia Sinica, Special Issue on Math,1 (1979),191-201 (in Chinese)] for determining the maximum non-orientable genus of a graph is developed to obtain the general theorem which presents a necessary and sufficient condition for a graph to be embeddable into either the orientable or the non-orientable surface of genus k. Furthermore,the greatest lower bound of the lengths of genus ranges of the class of nonplanar graphs which are up-embeddable is also obtained.
基金supported by the National Natural Science Foundation of China(No.10631060)
文摘Call a periodic map h on the closed orientable surface Σg extendable if h extends to a periodic map over the pair(S3, Σg) for possible embeddings e : Σg→ S3. The authors determine the extendabilities for all periodical maps on Σ2. The results involve various orientation preserving/reversing behalves of the periodical maps on the pair(S3, Σg). To do this the authors first list all periodic maps on Σ2, and indeed the authors exhibit each of them as a composition of primary and explicit symmetries, like rotations, reflections and antipodal maps, which itself should be interesting. A by-product is that for each even g,the maximum order periodic map on Σg is extendable, which contrasts sharply with the situation in the orientation preserving category.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No. ZYGX2010J109)National Natural Science Foundation of China (Grant No. 11101068)the Sichuan Youth Science and Technology Foundation (Grant No. 2011JQ0003)
文摘A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.
文摘In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficiently small.
