A Non-Conventional Coloring of the Edges of a Graph

Coloring the nodes of a graph is a commonly used technique to speed up clique search algorithms. Coloring the edges of the graph as a preconditioning method can also be used to speed up computations. In this paper we will show that an unconventional coloring scheme of the edges leads to an NP-complete problem when one intends to determine the optimal number of colors.


Introduction
Let be a finite simple graph.This means that has finitely many nodes, that is, does not have any double edge or loop.In this special case an edge of can be identified with a two element subset of .As a consequence the set of edges of G is a family of two element subsets of .A subgraph of G is called a clique if two distinct nodes of are always adjacent in G .A clique with nodes is simply called a -clique.The number of nodes of a clique is sometimes referred as the size of the clique.A 1-clique in G is a vertex of .A 2-clique in is an edge in .A 3-clique in is sometimes referred as a triangle in .A -clique in is called a maximal clique if it is not a subgraph of any -clique in .A k-clique in is defined to be a maximum clique if does not contain any 1  -clique.The graph may have several maximum cliques.However, each maximum clique in has the same number of nodes.This well defined number is called the clique size of and it is denoted by .
Determining the size of a maximum clique in a given graph is an important problem in applied and pure discrete mathematics.A number of selected applications are presented in [1] and [2].It was recognized in [3] that the efficiency of the clique search algorithms can be crucially improved by coloring and rearranging the nodes of the tested graph.In [4] instead of coloring the nodes a technique based on dynamic programming was used successfully.In [5] another coloring idea was presented.This time the edges of the graph were colored.Numerical evidence shows that the edge coloring provides sharper upper bounds for the clique size than the node coloring.
However, the improved upper bound comes for a higher computational cost.
We color the nodes of a finite simple graph

 
G V,E  using colors.We assume that the coloring satisfies the following conditions: 1) Each node of receives exactly one color; 2) Adjacent nodes in never receive the same color.We will call this an type coloring of the nodes of with k colors.The letter refers to the expression legal coloring.We may use the numbers as colors.The coloring can be defined by a map imply that v 1 and v 2 are not adjacent for each 1 2 V  .We color the edges of a finite simple graph   G V,E  using colors.We suppose that for the coloring the next two conditions hold: are not nodes of a 4-clique in Given a finite simple graph and given a positive integer .Decide if the edges of have type coloring using k colors.
The smaller is the number of colors for which the edges of have a type coloring the more useable is the coloring in connection with clique search.The main

The Auxiliary Graph H
In this section we construct an auxiliary graph.This will play the role of building blocks in further constructions.
Let us consider the graph given by its adjacency matrix in H is given in Figure 1.Right below each node we recorded the color of the node.But at this moment the reader may ignore the colors.In order to avoid a cluttered figure we used two copies of the nodes 1 and 14 .One can imagine that the figure is drawn on a strip of paper.Then we fold the strip to form a cylinder identifying the shorter sides of the strip.Thus the graph H is drawn on the surface of a cylinder and we arrange things such that the two copies of coincide 1 v Table 1.The adjacency matrix of graph H.This bullet represents the edge  1 2 .Scanning the rows and of the adjacency matrix we can see that 2 .This means that the edge 1 2 cannot be a side of any triangle in Then repeat the argument for all the 24 edges of H .
In order to prove the statement (2) let us assume on the contrary that there is an L type coloring . We may assume that and since this is only a matter of rearranging the colors 1, 2, 3 among each other.Note that . This portion of the reasoning can be followed in Figure 1.We distinguish four cases listed in Table 3.Let us consider case 1.As must hold.But 12 v and 13 are adjacent nodes in and so we get the contradiction that 12 13 Let us consider case 2. From and , it follows that . Each of the remaining cases can be handled in an analogous way.The reasoning can be followed in Figure 3.
The statement (3) can be proved by exhibiting a required coloring.This is done in Figure 4.

The Auxiliary Graph K
Using the graph H we construct a new graph K .Let , 1 1 4 be pair-wise distinct points.These will be the nodes of .We connect the nodes x and j y to one point the edges (1) collapse to an edge and from we get back an isomor-  phic copy of H .We summarize the properties of K we will need later in two propositions.
In each 4-clique in K there are exactly two primary edges.Proof.Clearly, K contains a 4-clique.In order to prove statement (1) it is enough to verify that K does not contain any 5-clique.We assume on the contrary that K contains a 5-clique.As the primary edges form a matching in K , a 5-clique in K can have only 0, 1, 2 primary edges.The cases are depicted in Figures 6-8, Figure 6.A 5-clique without primary edge.respectively.The primary edges are marked by double lines.Let us suppose first that the 5-clique does not have any primary edge.Then there are 5 primary edges joining to the 5-clique.From this it follows that H must contain a 5-clique.When the 5-clique contains exactly one primary edge, then the graph H must contain a 4-clique.Finally, when the 5-clique contains exactly two primary edges, then H must contain a 3-clique.But by Proposition 1, H does not contain any 3-clique.This contradiction proves statement (1) g y y to be equal to   j f v .A routine inspection shows that the edges of the 4-clique whose nodes are i i j j , , , have a type coloring with 3 colors.(The reader will notice that in fact we used only 2 colors to color the nodes of the 4-clique.)This coloring procedure can be repeated for each 4-clique in which has exactly two primary edges.By Proposition 2, each 4-clique in has exactly two primary edges and so each edge of is colored.Therefore the edges of have a type coloring with 3 colors.This proves statement (1).
In order to prove statement (2) let be a type coloring of the edges of K. Using g we can construct an L type coloring

The Auxiliary Graph
Figure 9 illustrates the construction.These edges connect the graphs and K  .The resulting graph is de- edges.The properties of the graph we will need later are spelled out in the next , x y and   14 , 14 x y must receive the same color.Because the colors of the edges can be exchanged among each other freely we may in fact prescribe the colors of these edges.Similarly, the edges of K  have a type coloring with 3 colors.In the coloring of the edges of B K  the colors of the edges y   must be equal.Again the color of these edges can be prescribed.Because of the presence of the edges (2) the edges  14 14 ,  x y and

The Main Result
We are ready to prove the main result of this paper.

B L
Let be all the vertices of G.This means that . We assign two points i and i to the node for each . We choose the points 1 , 1 to be pair-wise distinct.We connect the nodes i and .In other words In order to pro e the claim let us assume on the contrary that     . By P position 4, On the other hand the definitio h rin es t n of t e colo hat To complete e sh at the problem of deciding if the nodes a given simple graph have a L type coloring with 3 colors is an NPcomplete problem.Proofs of this well-known result can be found for example in [6,7].

 
, G V E  be a finite sim that find a B type coloring of the edges of G .A possible interpretation of the main result of this per is that determining the optimal number of colors is a computationally demanding problem.In practical computations we should be content with suboptimal values of the number of colors.

Figure 5 .
Figure 5.The correspondence between H and K.
remaining four edges of the 4-clique we color in the following way.Set g x y ,

Figure 9 .
Figure 9. Connecting the graphs K and K′.
a finite simple graph.Using we construct a new graph which satisfies the following two requirements: 1) If the nodes of have an type coloring using 3 colors, then the edges of type coloring with 3 colors, then the nodes of G have an type coloring with 3 colors.

Proposition 5 .
v are not nodes of a 4-clique in .In the lack of established terminalogy we call  the derived graph of G.The essential connection between G and G  is the following result.The nodes of have an type coloring with k colors if and only if the edges of have a type coloring with k colors.L G BThe reader will not have any difficulty to check the veracity of the proposition.The result makes possible to apply all the greedy coloring techniques developed for coloring the nodes of a graph.When the graph has n nodes it may have G instance when has 4000 nodes, then may have 10 millions edges.In this case the adjacency matrix of does not fit into the memory of an ordinary computer.Thus one should compute the entries of the adjacency matrix from the adjacency matrix of during the coloring algorithm.The most commonly used greedy coloring of the nodes of a graph takes this technique to the derived graph we get an algorithm whose computational complexity is  4O n .The author has carried out a large scale numerical experiment with this algorithm.The results are encouraging.

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The graph H has 14 nodes