# Coloring Number Of Planar Graphs

### Kuratowski s and wagner s theorems the polish mathematician kazimierz kuratowski provided a characterization of planar graphs in terms of forbidden graphs now known as kuratowski s theorem.

Coloring number of planar graphs. We show that the game coloring number of a planar graph is at most 19. Since the acyclic chromatic number of a planar graph is at most 5 it follows. A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph k 5 or the complete bipartite graph k 3 3 utility graph. 4 color theorem the chromatic number of a planar graph is no greater than 4 example 1 what is the.

Since a vertex with a loop i e. This is a special case of baranyai s theorem soifer 2008 provides the following geometric construction of a coloring in this case. This implies that the game chromatic number of a planar graph is at most 19 which improves the previous known upper bound for the game chromatic number of planar graphs. Vertex coloring when used without any qualification a coloring of a graph is almost always a proper vertex coloring namely a labeling of the graph s vertices with colors such that no two vertices sharing the same edge have the same color.

A connection directly back to itself could never be properly colored it is understood that graphs in this context are. This problem was first posed in the nineteenth century and it was quickly conjectured that in all cases four colors suffice. The coloring game on planar graphs was invented by steven j. Number of planar graphs is posed in 15 also see 14.

Recently xuding zhu made a. The game chromatic number of planar graphs was first studied by keirstead and trotter 3. Coloring 1 planar graphs were first studied by ringel 1965 who showed that they can be colored with at most seven colors. This was finally proved in.

If both the vertices and the faces of a planar graph are colored in such a way that no two adjacent vertices faces or vertex face pair have the same color then again at most six colors are needed borodin 1984. A mÃ¶bius strip requires six colors tietze 1910 as do 1 planar graphs graphs drawn with at most one simple crossing per edge borodin 1984. Place n points at the vertices and center of a regular n 1 sided polygon sided polygon. A complete graph k n with n vertices is edge colorable with n 1 colors when n is an even number.

In 7 dinski and zhu proved that if a graph has acyclic chromatic number k then its game chromatic number is at most k k 1. Brams and was published by gardner 2. The example of the complete graph k 6 which is 1 planar shows that 1 planar graphs may sometimes require six colors.

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