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Facial achromatic number of triangulations on the sphere (Women in Mathematics)
A graph consists of a set of vertices and a set of edges. A coloring of a graph is an assigning of colors to the vertices such that any adjacent vertices receive different colors. In particular, a coloring is called complete if every pair of colors appear on some edge. In this talk, we expand complete colorings of graphs to those of graphs embedded on surfaces and consider such colorings of even triangulations on the sphere
Asymmetric -colorings of graphs
We show that the edges of every 3-connected planar graph except can be
colored with two colors in such a way that the graph has no color preserving
automorphisms. Also, we characterize all graphs which have the property that
their edges can be -colored so that no matter how the graph is embedded in
any orientable surface, there is no homeomorphism of the surface which induces
a non-trivial color preserving automorphism of the graph
Three-coloring triangle-free graphs on surfaces III. Graphs of girth five
We show that the size of a 4-critical graph of girth at least five is bounded
by a linear function of its genus. This strengthens the previous bound on the
size of such graphs given by Thomassen. It also serves as the basic case for
the description of the structure of 4-critical triangle-free graphs embedded in
a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark
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