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The Connectivity and Minimum Degree of Circuit Graphs of Matroids

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Acta Mathematica Sinica,English Series 2010年第2期26卷 353-360页

作者： Ping LI gui zhen liu ematics and System Sciences Shandong University Ji'nan 250100 P. R. China School of Mathematics and System Sciences Shandong University Ji'nan 250100 P. R. China

Let G be the circuit graph of any connected matroid M with minimum degree 5（G）. It is proved that its connectivity κ（G） ≥2｜E（M） - B（M）｜ - 2. Therefore 5（G） ≥ 2｜E（M） - B（M）｜ - 2 and this bound is t... 详细信息

关键词： matroid circuit graph of matroid connectivity

f-Colorings of Some Graphs of f-Class 1

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Acta Mathematica Sinica,English Series 2008年第5期24卷 743-748页

作者： Xia ZHANG gui zhen liu College of Mathematics Science Shandong Normal University Ji'nan 250014 P. R. China School of Mathematics and System Sciences Shandong University Ji'nan 250100 P. R. China School of Mathematics and System Sciences Shandong University Ji'nan 250100 P. R. China

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v V（G） at most f（v） times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by X′f（G）. Any simple graph G has the f-chromatic index equal to △f（G） or △f（G）＋ 1, where △f（G） =max v V（G）{[d（v）/f（v）]}. If X′f（G） = △f（G）, then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with △f（G） colors are given.

关键词： simple graph edge-coloring f-coloring classification of graphs f-chromatic index

On 2-Factors with Prescribed Properties in a Bipartite Graph

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Acta Mathematica Sinica,English Series 2006年第4期22卷 1115-1120页

作者： Jin YAN gui zhen liu School of Mathematics and Systems Science Shandong University Jinan 250100 P. R. China

liu and Yan gave the degree condition for a balanced bipartite graph G = （V1, V2; E） to have k vertex-disjoint quadrilaterals containing any given k independent edges e1,……, ek of G, respectively. They also conjectured that for any k independent edges e1,……, ek of G, G has a 2-factor with k cycles C1, C2, ……, Ck with respect to {e1, e2,……, ek} such that k - 1 of them are quadrilaterals. In this paper, we prove this conjecture.

关键词： Bipartite graph Vertex-disjoint Quadrilateral 2-Factor

Group Edge Choosability of Planar Graphs without Adjacent Short Cycles

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Acta Mathematica Sinica,English Series 2013年第11期29卷 2079-2086页

作者： Xin ZHANG gui zhen liu Department of Mathematics Xidian University School of Mathematics Shandong University

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 △（... 详细信息

关键词： Group edge coloring list coloring planar graphs short cycles girth

On Edge Colorings of 1-Toroidal Graphs

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Acta Mathematica Sinica,English Series 2013年第7期29卷 1421-1428页

作者： Xin ZHANG gui zhen liu Department of Mathematics Xidian University School of Mathematics Shandong University

A graph is 1-toroidal, if it can be embedded in the torus so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-toroidal graph with maximum degree △ ≥ 10 is of class one in terms of edge coloring. Meanwhile, we show that there exist class two 1-toroidal graphs with maximum degree △ for each A ≤ 8.

关键词： 1-Toroidal graph 1-planar graph edge coloring

Every Toroidal Graph Is Acyclically 8-Choosable

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Acta Mathematica Sinica,English Series 2014年第2期30卷 343-352页

作者： Jian Feng HOU gui zhen liu Center for Discrete Mathematics Fuzhou University School of Mathematics Shandong University

A proper coloring of a graphG is acyclic if G contains no 2-colored cycle.A graph G is acyclically L-list colorable if for a given list assignment L={L（v）：v∈V（G）},there exists a proper acyclic coloringφof G such thatφ（v）∈L（v）for all v∈V（G）.If G is acyclically L-list colorable for any list assignment L with｜L（v）｜≥k for all v∈V（G）,then G is acyclically *** this article,we prove that every toroidal graph is acyclically 8-choosable.

关键词： Acyclic coloring choosability toroidal graph

k-Factors in Regular Graphs

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Acta Mathematica Sinica,English Series 2008年第7期24卷 1213-1220页

作者： Wai Chee SHIU gui zhen liu Department of Mathematics Hong Kong Baptist University Hong Kong P. R. China School of Mathematics and Systems Science Shandong University Ji'nan P. R. China

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.

关键词： regular graph k-factor edge-connectivity

List Total Colorings of Planar Graphs without Triangles at Small Distance

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Acta Mathematica Sinica,English Series 2011年第12期27卷 2437-2444页

作者： Bin liu Jian Feng HOU gui zhen liu Department of Mathematics Ocean University of China Qingdao 266100 P. R. China Center for Discrete Mathematics Fuzhou University Fuzhou 350002 P. R. China School of Mathematics Shandong University Ji＇nan 250100 P. R. China

Suppose that G is a planar graph with maximum degree △. In this paper it is proved that G is total-（△ ＋ 2）-choosable if （1） △ ≥ 7 and G has no adjacent triangles （i.e., no two triangles are incident with a common edge）; or （2） △ ≥6 and G has no intersecting triangles （i.e., no two triangles are incident with a common vertex）; or （3） △ ≥ 5, G has no adjacent triangles and G has no k-cycles for some integer k ∈ {5, 6}.

关键词： List total coloring choosability planar graph

Light Subgraphs in the Family of 1-Planar Graphs with High Minimum Degree

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Acta Mathematica Sinica,English Series 2012年第6期28卷 1155-1168页

作者： Xin ZHANG gui zhen liu Jian Liang WU School of Mathematics Shandong University

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2 V （K1 ∪ K2） with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1 V （K1∪K2） with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

关键词： 1-Planar graph lightness height discharging