Wir stellen einen persönlichen Weg vor, um die Suche nach einer chromatischen Zahl eines beliebigen Graphen auf die Suche nach dem chromatischen Index eines 3-regulären Graphen mit der Bezeichnung kubisch ohne Isthmus zu transformieren.We present a personal transformation of the chromatic number research in any graph into chromatic index research in bridgeless cubic graph.Nous présentons une manière personnelle de transformer la recherche d'un nombre chromatique d'un graphe quelconque en la recherche de l'indice chromatique d'un graphe 3-régulier dit cubique, sans isthme

cousin, xavier

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To transform the chromatic number research to the
chromatic index one in cubic graph
Xavier Cousin
To cite this version:
Xavier Cousin. To transform the chromatic number research to the chromatic index one in cubic
graph. 2019. ￿hal-02334852v2￿
Xavier Cousin   1/3 
To transform the chromatic number search to the chromatic index one in cubic graph 
 
 
Summary: We present a personal transformation of the chromatic number research in any graph into chromatic 
index research in bridgeless cubic graph.  
 
Keywords:  Chromatic number, chromatic index, edge coloring, vertices coloring, graph theory 
 
 
 
Résumé : Nous présentons une manière personnelle de transformer la recherche d’un nombre chromatique d’un 
graphe quelconque en la recherche de l’indice chromatique d’un graphe 3-régulier dit cubique, sans isthme. 
 
Mots Clefs : Nombre chromatique,  indice chromatique, coloration d’arêtes, coloration de sommets, théorie de 
graphes. 
 
 
 
 
 
1. Introduction 
 
We first remind some notions and naming convention and particularly the Cartesian sum of graph. 
 
Knowing an algorithm for edges colouring of a 3-regular graph called cubic, without isthmus, we know how to 
find the chromatic number of any graph. Indeed, we explain how to reduce it to a problem SAT, then 3-SAT and 
following the article of Ian Holyer [3], we reduce it to a problem of chromatic index research of a cubic graph 
without isthmus. 
 
 
2. Reminders and conventions 
 
• G = (X, A) is the graph which has X as set of vertices and A as set of edges. 
• |Y| is the cardinal of a set Y. 
• We note dG(x) the degree of x vertex on graph G. 
• A vertex clique is a set of vertices linked together in pairs by edges. We note   the maximum number of a 
clique. 
• N is the vertices number of G (or |X|) and M the edges number (or |A|). 
• We name χ’, the chromatic index of G corresponding to the minimum number of colours to strongly colour the 
edges. In the books of Claude Berge [1] [2], this number is noted q(G). 
• χ is the chromatic number of G corresponding to the minimum number of colours to strongly 
colour the vertices. In the books of Claude Berge [1] [2], this number is noted  (G). 
• Kq represents the order clique q. 
• G  is noted as the complement of G. 
• G1 + G2 denotes the Cartesian sum of two graphs (cf. [1] [2]). 
 
 
Reminder: Cartesian sum of graph. A Cartesian sum G
e
 of a graph G with another graph R is a graph consisting 
of the union of k duplications of the graph G, where k is the cardinal of XR and edges of R connecting the 
vertices of the different duplicates (Gi). We can see graphically a duplication Gi in a horizontal axis and a 
duplication Rj in the vertical plane. 
Here we will mainly use the Cartesian sum G + Kq. 
 
 
Example: Let a graph G with X = {pa, pb, ca, cb} and A= {(pa, ca), (pa, cb), (pb, ca)},  
 
G
e
 = G + K3 = {(pa1, pa2), (pa1, pa3), (pa2, pa3), (pb1, pb2), (pb1, pb3), (pb2, pb3), (ca1, ca2), (ca1, ca3), (ca2, 
ca3), (cb1, cb2), (cb1, cb3), (cb2, cb3)}).  
 
 
 
 
Xavier Cousin   2/3 
It is the Cartesian sum, G + K3 which gives graphically: 
 
 
pa pb ca cb 
G 
pa1 pb1 cb1 ca1 G1 
pa2 
pb2 
cb2 ca2 
G2 
pa3 pb3 cb3 ca3 
G3 edges of K3 
linked to cb 
G e 
 
Resulting graph 
 
The edges set {(cb1, cb2), (cb1, cb3), (cb2, cb3)} will be named K3(cb) linked to the vertex cb of G. 
 
Remark: for the chromatic number research χ, we will associate the colour i to the ith duplication. 
 
 
3. From chromatic number to chromatic index 
 
This part takes up the graph sum Cartesian G + Kq and is based on the work of Ian Holyer seen in [3]. We will 
show that looking for the chromatic number of G is like developing G in several steps: first in a graph G+Kq. 
Once done, the second step is to create a problem of SAT who will be in third turned into a problem 3-SAT. The 
fourth step resulting from the work of Ian Holyer [3] will apply a transformation of the problem 3-SAT into a 
cubic graph where it will only remain to apply chromatic index research. 
 
Remark: Garey et Johnson [4][5] have already shown how to reduce a vertex coloring (chromatic number) 
problem into a 3-SAT. The following steps present a personal version. 
 
Stage 1. Let us take again the original graph G. We transform it into a graph G+Kq with Gi = (Xi, Ai) the graph 
of ith iteration where vertices xi will mean « x is of colour i » and with Kq(x) the clique of cardinality q and 
linked to the vertices x, linking then all vertices xi, i=1 to q. We can initialize by default q as being the maximum 
degree. 
 
Stage 2. We will transform this resulting graph into a problem to satisfy (SAT in the sense of Garey and Johnson 
[4][5]) with clauses et literals. These problems have boolean variables xi used in literals (xi or not xi noted xi). 
These literals are linked by the operator  « or » noted  to create clauses which are linked between them by the 
operator « and » noted ˄. Thus the transformation takes each vertex xi in litteral and puts three types of clauses 
which are: 
 
 On a same Gi=(Xi, Ai), if (xi, yi) belongs to Ai, then the following clause is: (not xi or not yi) or (xi  
yi) 
o This indicates that two vertices connected by an edge cannot have the same colour i but cannot 
be coloured by this colour i (both variables are 'false') 
 On a same Kq(x), for all xi of Gi and all xj of Gj, (not xi or not xj)  or (xi  xj) 
o This indicates that the same vertex cannot cumulatively have two colours i and j 
 On a same Kq(x) of cardinality q, (x1 or x2 or … or xq) or (x1  x2  …  xq) 
o This indicates that there is at least one colour between 1 and q for the same vertex x 
 
Stage 3. These clauses are then retransformed into 3-SAT: clauses with two literals have one more literal which 
is always false and the clause with n literals is transformed by the intermediate literal method (a or b or c or d or 
e) = (a or b or int1) and (no int1 or c or int2) and (not int2 or d or e). Note this transformation 3-SAT(G).  
 
Stage 4. We apply the Ian Holyer [3] method. We note 3Sat-3Reg (problem 3-SAT) the method of creating a 3-
regular graph for the search of its chromatic index from a problem 3-SAT. If the graph of degree 3 resulting from 
this method is 4 colourable then the problem 3-SAT is not satisfied. If it is 3-colourable, then the 3-SAT problem 
is satisfied giving the colouring of the graph G. 
 
Iteration on the four stages. We loop from i = q to 1, starting from graphs G + Ki starting with a 'q' close to an 
upper bound (max degree for example). If the 3-SAT problem coming from the graph G+Ki is satisfied but the 
graph G + Ki-1 is not satisfied then the chromatic is equal to i. 
 
Xavier Cousin   3/3 
 
4.  Synthesis 
 
Here is a summary of the main results with G=(X,A) and G+Kq =     
 
               where Gi =(Xi, Ai) i
th
 
copy of G and Kq(x) copy of Kq linked to a vertex x of X : 
 
 χ to χ’ ; Let SATi=   
{ 
                    
          
           
       ˄ 
                                 ˄ 
                    
} 
 With the two transformations, 3Sat-3Reg to transform a 3-SAT problem to a cubic graph and 3-SAT the 
transformation of a NSAT problem into a 3SAT one, we have: 
 i 1  i  upper bound of χ (max degree for example), such that χ’(3Sat-3Reg (3-SAT(SATi)))=3 and 
χ’(3Sat-3Reg (3-SAT(SATi-1)))=4 and χ(G)=i. 
 
 
5. Conclusion 
 
We describe how one can reduce the search for the chromatic number of any graph to the search of the chromatic 
index of a particular cubic graph by way of transformations into problems SAT. 
 
We will show in further research how to carry out this search for the chromatic index of a cubic graph. 
 
We will show in future article the use of these algorithms and a particular construction of graph to solve some 
problems of operational research as scheduling of tasks and even of allocation of resources. This article will 
explain how to set at the same time constraints of order, disjunction and even cumulative "a minima" or "a 
maxima" in order to optimize these resources. 
 
6. References  
 
[1] C. Berge, Graphes, Bordas, Paris, 1983. 
[2] C. Berge, Hypergraphes, Combinatoire des ensembles finis, Bordas, Paris, 1987. 
[3] Ian Holyer, The NP-Completeness of edge-colouring, SIAM J. Comput, Vol 10, N° 4, November 1981 
pp 718-720. https://pdfs.semanticscholar.org/37c9/2599655b9f66850f1e4e0e48ad045abad8f6.pdf 
[4] M. R. Garey, D. S. Johnson, L. Stockmeyer , Some simplified polynomial complete problems, Proc. 6th 
Annual ACM Symposium on Theory of computing, pp 47-63, 1974. 
[5] M. R. Garey, D. S. Johnson, Computers and Intractibility, A guide to the Theory of NP-Completeness, 
W. H. Freeman and Company, San Franscisco, 1979. 


Comment transformer la recherche du nombre chromatique d'un graphe quelconque en celle de l'indice chromatique d'un graphe cubique

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Comment transformer la recherche du nombre chromatique d'un graphe quelconque en celle de l'indice chromatique d'un graphe cubique

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