To find more 3-uncolorable vertices recognizable in polynomial time, we consider mobile 3-uncolorable vertices. In color space, the transfer is expressed by shift of dot strings in alternate configurations. An alternate configuration is the union of a DJC and movable dot strings relative to the DJC. It has several FLC one of which is a WFLC. The movement of the movable dot string exchanges FLC with WFLC to ensure that WFLC exists always in the configuration. We prove that an additional dot string (ADS) to an alternate configuration may turn the configuration to be a triplet. So, 3-uncolorable vertices in the graph of alternate configurations in planar graph with degree more than 4 G(V,E) can be recognized in O(|V8|). We need to find more recognizable 3-uncolorable vertices in polynomial time for judging 3-coloring of planar graph.

How to build the arithmetic finding 3-uncolorable vertices in polynomial time in 3-coloring is studied. For planar graph G[V,E] the diagrammatic arithmetic computing jammed configurationwith the independent clip (DJC) is built in the color space according to the Grötzseh theorem. Each DJC corresponds a triplet in G[V,E] , which is the invariant set of three sets of monochromatic vertices with different colors under coloring transformation. The algorithmcancomplete the computation to found all DJC in 8 O(V ) . The vertex adjoins vertices of three monochromatic sets is 3-uncolorable. They are only the part of 3-uncolorable vertices. So, 3-coloring has 3-uncolorable vertices can be recognized in polynomial time.

Color space is introduced to discuss 3-coloring of a graph G(V,E). It is a 3x|V| lattice that every column expresses a vertex and every row expresses a color. A color space can provide with all the information of a colored graph that include the topology, coloring and coloring transformation. So, it reduces calculation in searching 3-uncolorable vertices, And, we may try to find a deterministic algorithm to search some 3-uncolorable by using the color space.

Within the color space of 3-coloring, we build deterministic algorithms for plane graphs with no more than 4 to find base configurations that appear in the n-DJC group, and thus find new triplets bringing 3-inseparable vertices. This paper provides the basic theory of the n-DJC group, including the removable conflict group between DJC and the rotation of DJC in the group. We can identify 3-uncolorable vertices according to unreasonable removable conflicts or frozen rotations. And we point out that such n-DJC groups can appear in the color space of planar graph with degree no more than 4.