孤独乱语

Xserver配置Mathematica字体

“南开之星”上安装了Mathematica 5.2的并行版本。在远程启动的时候总会报字体错误，然后工具栏上的好多数学符号都不能显示。

解决方法是：到Mathematica官方网站下载MathFonts字体。具体下载路径是：
http://support.wolfram.com/mathematica/systems/linux/general/latestfonts.html

将字体文件解压以后进行安装。

对于Windows系统，一般使用的Xserver有Xwindow, 将解压的字体复制到Xwindow目录下面的fonts目录下，并修改Xwindow的字体路径设置。便可以正常使用该字体了。其他Windows下面的X服务基本上应该一样。我用的Xming只需要将字体复制到目录下面，不需要进行任何设置便可以正常工作。

对于Linux系统，则需要设置FontPath。
如果临时使用的话，登录以后，在终端运行
xset fp+ 解压的字体路径／Type1
即可在本次登录内正常使用该字体。

现在一般都使用Xorg了，可以直接修改/etc/X11/xorg.conf
添加类似下面的一行
FontPath "/usr/share/fonts/MathFonts/Type1"
路径根据字体所在路径设置。

注意一定要设置到Type1路径，如果只设置到上级目录，虽然xorg可以找到，但是xset 却显示没有找到路径。

Total Coloring

In graph theory, total coloring is a type of coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be proper in the sense that no adjacent vertices, no incident edges, and no edge and its endvertices are assigned the same color. The total chromatic number χ″(G) a graph G is the least number of colors needed in any total coloring of G.

The total graph T = T(G) of a graph G is a graph such that (i) the vertex set of T corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T if and only if their corresponding elements are either adjacent or incident in G. Then total coloring becomes a (proper) vertex coloring of the total graph.

Some properties of χ″(G):

1. χ″(G) ≥ Δ(G) + 1.
2. χ″(G) ≤ Δ(G) + 1026. (Molloy, Reed 1998)
3. χ″(G) ≤ Δ(G) + 8 ln8(Δ(G)) for sufficiently large Δ(G). (Hind, Molloy, Reed 1998)
4. χ″(G) ≤ ch′(G) + 2.

Here Δ(G) is the maximum degree; and ch′(G), the edge choosability.

Total coloring arises naturally since it is simply a mix of vertex and edge colorings. The next step is to look for any Brooks-typed or Vizing-typed upper bound on the total chromatic number in terms of maximum degree. It turns out that the total coloring version of maximum degree upper bound is a difficult problem and has eluded mathematicians for 40 years. The most well-known speculation is the following.

Total coloring conjecture. (Behzad, Vizing)

χ″(G) ≤ Δ(G) + 2.

Apparently, the term "total coloring" and the statement of total coloring conjecture were independently introduced by Behzad and Vizing in numerous occasions between 1964 and 1968. See [Jensen, Toft 1995] for details. The conjecture is known to hold for a few important classes of graphs, such as all bipartite graphs and most planar graphs except those with maximum degree 6. The planar case can be completed if Vizing's planar graph conjecture is true. Also, if the list coloring conjecture is true, then χ″(G) ≤ Δ(G) + 3.

Results related to total coloring have been obtained. For example, Kilakos and Reed (1993) proved that the fractional chromatic number of the total graph of a graph G is at most Δ(G) + 2.

References

* Hind, Hugh; Molloy, Michael; Reed, Bruce (1998). Total coloring with Δ + poly(logΔ) colors. SIAM J. Comput. 28(3), 816–821.
* Jensen, Tommy R.; Toft, Bjarne (1995). Graph coloring problems. New York: Wiley-Interscience. ISBN 0-471-02865-7.
* Kilakos, Kyriakos; Reed, Bruce (1993). Fractionally colouring total graphs. Combinatorica 13, 435–440.
* Molloy, Michael; Reed, Bruce (1998). A bound on the total chromatic number. Combinatorica 18, 241–280.

研究图论的一个辅助工具

http://www.mi.sanu.ac.yu/newgraph/index.html

newGRAPH is a fully integrated environment used for improving a research
process in graph theory. Its purpose is:

• help a researcher pose, verify or disprove a conjecture
• experiment with graphs
• educative application

It is a new version of GRAPH, written by Dragoš Cvetković
and his collaborators. It is currently being developed by Dragan Stevanović
and Vladimir Brankov, with Dragoš Cvetković and Slobodan Simić
serving as consultants.

Resources Online about Mathematics

Mathematics Journals on the Web

Mathematics Archives on the Web

Social Network Theory

Social Network Theory
Theory Used

Graph Coloring Problems: the archives

http://www.imada.sdu.dk/Research/Graphcol/

Open Problems

Some Problems in Matroid Theory by Thomas Zaslavsky

http://www.math.binghamton.edu/zaslav/Matroids/matroidprobs.html

Problem of the month by Bojan Mohar

http://www.fmf.uni-lj.si/~mohar/Problems.html

PERFECT PROBLEMS by Vašek Chvátal
http://www.cs.concordia.ca/~chvatal/perfect/problems.html

Problems in Topological Graph Theory by Dan Archdeacon

http://www.emba.uvm.edu/~archdeac/problems/problems.html

Open Problems - Graph Theory and Combinatorics by Douglas B. West

http://www.math.uiuc.edu/~west/openp/