On a Sufficient and Necessary Condition for Graph Coloring

Using the linear space over the binary field that related to a graph G, a sufficient and necessary condition for the chromatic number of G is obtained.


Let ( )
, G V E = be a graph, where V is a set of vertices and E is a set of edges of G.A vertex coloring of a graph G is a coloring to all the vertices of G with p colors so that no two adjacent vertices have the same color.Such the graph is called p -coloring.The minimal number p is called the chromatic number of G, and is de- noted by ( ) G χ .The so-called Four Color Problem is that for any plane graph G, ( ) 4 G χ ≤ [1].The coloring of a graph G is an interesting problem for many people [2].This is mainly caused by the Four Color Problem [3].
In this paper, putting a graph into a linear space over the binary field ( ) , we obtain the sufficient and necessary condition for the chromatic number of G.
And as an application of above result, we give a characterization for a maximal plane graph to be 4-coloring.

The Linear Space A n over GF(2)
Now we introduce the linear space over the field ( ) . Firstly, the field ( ) contains only two members: , where the addition and multiplication are as usual excepting that 1 1 0 , , ,  be the n vertices, the all vectors of the linear space n A are formed of the symbolic expression It has 2 n vectors.The addition of two vectors is defined by ( ) Here, the n vertices { } , , , n a a a  will serve as the most basic elements of the linear space n A .They will be as a basis of the linear space n A .For them the basic assumption is that these n vertices are linearly indepen- dent in n A .0 u u + =.
Here we denote the zero vector by 0.
For a vector 1 , the order of the vector u is defined by where the ' ∑ means that the addition is the usual addition in the integer set.
A vector with k order is called a k-order vector, and a vector whose order is even is called an even-order vector.
We now give some structures to the linear space n A .In other words, we want to "put" graph into the linear space n A .In the linear space n A , 1-order vectors are vertices of a graph.The edge is expressed as the 2-order vertex, i.e.
i j a a + is the edge i j a a .So we have two ways to describe a edge: by i j a a (in the usual sense), or by i j a a + (in the linear space n A ).In the following, we always discuss a graph in the linear space n A , it means we express edges with the second form.
All the 2-order vectors in the linear space n A are the all possible edges with n vertices { } , , , n a a a  , that we denote by A E : For a giving graph ( ) with n vertices, in the linear space n A , the elements of the set E are the 2-order vectors of n A , then the edge set E of G is the subset of the set A E , A E E ⊆ .We give two examples here.1) For the set A E with all the 2-order vertices in n A , the graph ( ) is a complete graph, whose any two vertices are adjacent.
2) For a graph ( ) with n vertices, the complementary set of E in the set of the 2-order vertices We now see the addition in n A .For a path of G with a sequence of edges , where the end-points are a and b , the sum of the edges is: This expression indicates the relation between the addition in the linear space and the connectivity of a graph.That is why we put the graph into the linear space n A .Lemma 1.The sum of even-order vectors is even-order.This is clear by the property that 0 i j a a + = if and only if i j = .As a special case of the Lemma 1, we have Lemma 2. Let ( ) then k is even.Definition.Let A n be the n-dimensional linear space derived by the graph ( ) then G R is called an outer-kernel subspace of G .And G R is a maximal outer-kernel subspace if the rank of G R is maxima in all the outer-kernel subspace of G .Now we give some basic properties of a outer-kernel subspace of a graph G .By definition, G R is a subspace of n A .Denote the set of all 2-order vectors of G R by ( ) , , . Hence all the 2-order vectors formed by the set of vertices H span the linear subspace B of G R .Thus the connected block 1 , then k is even and there exists a

The Main Results
The outer-kernel subspace plays an important role in the problem of vertex coloring.Theorem 1.Let n G be a graph with n vertices, then the sufficient and necessary condition for n G to be p-coloring is that the rank of an outer-kernel subspace G R of n G is n p − .Proof.First we prove the necessity.Suppose that the graph n G is p -coloring.Then all the vertices of n G can be divided into p subsets by the colors: , , , , , , , That means the vertices with a same color are in the same subset.Because it may have a subset with only one vertex, we denote the one-vertex subsets with different colors by 1 , , . The elements of subset ( ) . For the sufficiency, suppose that there exists an outer-kernel subspace G R with condition (1), and the dimension of G R is n p − .We divide the vertices of G into some subsets according to the subspace G R .If for two vertices a and b there have then we put a and b into a same subset.Like the notation of congruence we denote ( ) Obviously, if a vertex a appears in G R , then there has at least another vertex in the same subset with a.If a vertex does not appear in G R , then this vertex forms a subset by itself, i.e. the subset contains only one vertex.Lemma 5.The vertices from different subset are linear independence on G R , i.e. if 1 2 , , , m a a a  belong to different subsets respectively, then ( ) a a is in the same subset.Now we go on with the proof of the sufficiency.Suppose that the 2-order vectors 1 2 , , , n p r r r −  form a basis of G R , and the vertices of the graph G are now divided into the disjoint subset 1 2 , , , l N N  by the method above.Take  , then any vertex a of G must be in some subset i N and by (5) we have So any vertex of G can be expressed by i b and an element of G R .Thus by Lemma 5, A .Hence l p = .By the definition of G R and (5) we know that the two vertices in the same subset i N are non-adjacent.Thus, we can assign one color to the vertices of each subset i N .So we just need p colors for G.The graph is p -coloring.
Due to Theorem 1 and the expression (4), we have: Theorem 2. For a graph G with n vertices, the sufficient and necessary condition for ( )

An Application to Plane Graphs
As an application of Theorem 1, we consider a result of the coloring to the plane graph.
A maximal plane graph is a graph G such that for any two non-adjacent vertices a and b of G, G added to the edge ab makes a non-planar graph.It is clear that all the faces of a maximal plane graph are triangles.
A maximal plane graph is 3-CR-edge coloring if we can color its edges by 3 colors such that the three edges of every its triangle face are coloring by different colors.Later we will see that the CR in the definition is borrowed from the Cauchy-Riemann condition in the complex function theory.
Theorem 3. If a maximal plane graph is 3-CR-edge coloring, then the graph is 4-vertex coloring.
The inverse of the Theorem 3 is true, too.That means if a maximal plane graph is 4-vertex coloring, then the graph is 3-CR-edge coloring.
Proof.We introduce the 2-demensional linear space 2 A : G V E = be 3-CR-edge coloring, and the all edges of G can map to the three elements ( ) ( ) ( ) 0,1 , 1,0 , 1,1 of 2 A by their colors, respectively.That is the mapping f For a path of G with the end-point a, b and the sequence of the edges By the condition of 3-CR-edge coloring, the extending mapping f by ( 7) is dependent only on the end-point a and b, and independent on their path.
Let n A′ be the ( ) 1 n − -dimensional linear subspace spanned by all the 2-order vectors of the space n A .Then f is the homomorphic mapping from subspace n A′ onto the space A 2 .The homomorphic kernel R con- sists of such vector e of n A′ that satisfies OPEN ACCESS OJDM ( ) 0 f e = (8) Suppose R is the subset of 2-order vectors of n A′ that satisfies (8) and R is spanned by R. Then by (8) we have Denote the linear independent spanning elements of R by 1 2 , , , m e e e  , that is just the basis of R .And dim R m = .We take , , e e e E of linear space n . By Theorem 1, the graph G is 4-vertex coloring.