We survey some geometric and analytic results under the assumptions of combinatorial curvature bounds for planar/semiplanar graphs and curvature dimension conditions for general graphs.
In this paper, we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group （△ （G）＋1）-edge-choosable, and some planar graphs with large girth and maximum degree are group △（...In this paper, we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group （△ （G）＋1）-edge-choosable, and some planar graphs with large girth and maximum degree are group △（G）-edge-choosable.展开更多
Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient condit...Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient conditions for a graph to have a vertex v such that d2（v） ≥ d（v）, where d（v） denotes the degree of v？ Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K4^--free graph or every maximal planar graph is shown to have a vertex v such that d2（v） ≥ d（v）. Other sufficient conditions on graphs for guaranteeing this property axe also proved.展开更多
基金This work was supported partially by the National Natural Science Foundation of China (Grant No.10471131)the Natural Science Foundation of Zhejiang Province(Grant No.Y604167)
文摘In this paper we prove that every planar graph without 4,6 and 8-cycles is 3-colorable.
基金partially supported by NSFC (No.12001296)Fundamental Research Funds for the Central Universities+3 种基金Nankai University (No.63201163)Shi is partially supported by NSFC (No.11922112)Natural Science Foundation of TianjinNankai Universitv (No.63206034)。
基金Acknowledgements B.H. was supported by the National Natural Science Foundation ot China （Grant No. 11401106） Y. L. was supported by the National Natural Science Foundation of China （Grant No. 11271011）, the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China （11XNI004）.
文摘We survey some geometric and analytic results under the assumptions of combinatorial curvature bounds for planar/semiplanar graphs and curvature dimension conditions for general graphs.
基金Supported by the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2013JQ1002)the Fundamental Research Funds for the Central Universities(Grant No.K5051370003)National Natural Science Foundation of China(Grant Nos.11101243,11201440,11301410 and 61070230)
文摘In this paper, we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group （△ （G）＋1）-edge-choosable, and some planar graphs with large girth and maximum degree are group △（G）-edge-choosable.
基金Supported by the Ministry of Education and Science,Spainthe European Regional Development Fund (ERDF)under project MTM2008-06620-C03-02+2 种基金the Catalan Government under project 2009 SGR 1298CONACyTMxico under project 57371PAPIIT-UNAM IN104609-3
文摘Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient conditions for a graph to have a vertex v such that d2（v） ≥ d（v）, where d（v） denotes the degree of v？ Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K4^--free graph or every maximal planar graph is shown to have a vertex v such that d2（v） ≥ d（v）. Other sufficient conditions on graphs for guaranteeing this property axe also proved.
