Wir beschreiben eine weitere Äquivalenz zwischen der Suche nach der Anzahl der Stabilität und der der chromatischen Zahl in beide Richtungen durch die Verwendung der kartesischen Summe von Graphen.We describe still another equivalence between stability number research and chromatic number research in both way by using a Cartesian sum of graphs.Nous décrivons une autre équivalence entre la recherche du nombre de stabilité avec celui du nombre chromatique dans les deux sens par l'usage de la somme cartésienne de graphes.

cousin, xavier

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Xavier Cousin   1/4 
Graph Cartesian sum: pivot between chromatic number and stability number 
 
 
 
 
Summary: We describe still another equivalence between stability number and chromatic number. We 
transform the concerned graph using a Cartesian sum and based on an algorithm to find the corresponding 
number.  
 
Keywords:  Chromatic number, stability number, graph theory 
 
 
 
Résumé : Nous décrivons une autre équivalence entre la recherche du nombre de stabilité avec celui du nombre 
chromatique dans les deux sens. Nous transformons le graphe concerné par l’usage de la somme cartésienne par 
l’utilisation d’un algorithme afin de trouver le nombre recherché. 
 
Mots Clefs : Nombre chromatique, nombre de stabilité, théorie de graphes. 
 
 
 
1. Introduction 
 
We first remind some notions and naming convention. It allows introducing the “Cartesian sum” of graphs. 
 
Then from any graph, knowing a research algorithm of the chromatic number, we deduce an algorithm to find 
the number of stability of this graph by building a particular graph based on Cartesian sum of the initial graph 
with claws. We show the opposite: from a known algorithm of the number of stability research, we deduce an 
algorithm to find the chromatic number of any graph. 
In literature, we find the way from stability number to chromatic number [1] but not necessarily the other way 
from chromatic number to stability number using the same mechanism. Here, we remind and clarify the both 
ways. 
 
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. 
• We name  the stability number; it is the maximum set of independent vertices (no edge between these 
vertices two by two). It's the opposite or the complement of the clique. 
• N is the vertices number of G (or |X|) and M the edges number (or |A|). 
• 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. 
• We note   the minimum number of cliques that partition all the vertices. 
• 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   2/4 
 
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. Equivalence between stability number and chromatic number 
We show how to use a supposedly known algorithm to find the chromatic number in another to compute the 
number of stability. We also explain the opposite, that is to say how to obtain the chromatic number knowing a 
method to find the number of stability. 
a) From stability number to chromatic number 
Theorem 4 : - to χ (Reminder [1]) For a simple graph G = (X, A), to search χ in a graph G = (X, A) is to look 
for the stability number iteratively in Cartesian sums Ge of graphs Ge = (G + Kq) on q going from an upper 
bound (max degree for example) to 1. 
In summary, we work iteratively q from upper bound to 1 by searching  (G+Kq). We find χG=i when i is 
such that (G+Ki-1)≠|X| and (G+Ki)=|X|. 
Proof: This theorem is is an augmented version of Berge's theorem and corollary ([1]). 
 
We want to find χ in a simple graph and we know a bound (at worst the maximum degree). 
We take a graph G
e
 = (G+Kq) with q initialised with an upper bound, q duplications of G and a set of edges such 
that all duplicated vertices xi (from 1 to q) coming from the same basis vertex x of G then linked two by two 
(cliques).  
To find a maximum stable set equal to in Ge indicates if there is or not a q-colouring (colouring to q colours) 
in G. If  = |X| then there is a q-colouring. 
The reader will verify for himself that it is impossible to have a  >|X| due to the cliques between vertices of Ge 
coming from the same base vertex. Of all the duplicate vertices from the same base vertex, only one will belong 
to the stable set. We can thus confuse the doubled vertex xi of G
e
 of the maximum stable set with the reference 
vertex, x of G. The vertices of this maximal stable set of G
e
 belonging to the same i
th
 duplication of G, will give 
the i
th
 colour to the vertices of corresponding reference. 
  
Xavier Cousin   3/4 
Example: Let's take the example of the figure of the reminder. The maximum degree is equal to 2, the upper 
bound of χ is 2. We thus have the graph resulting from the following figure with the search for a stable 
maximum set (here {cb1, ca1, pa2 , pb2}) and the induced colouring (ca and cb of the same colour  and pa, pb 
of another colour ): 
 
pa1 pb1 
cb1 
ca1 G1 
pa2 
pb2 
cb2 ca2 
G2 
G 
e 
 
Search of χ 
We go through the construction process of G
e
 with q = upper bound down to q = 1. We know that there exists q-
staining with q = upper bound, so  initial = |X|. As soon as current  courant <|X| of G then q-colouring is no 
longer possible. So the chromatic number χ of G is equal to q + 1 (q is the current value). 
b) From chromatic number to stability number 
Theorem 5 : - χ to  To search the stability number in a simple graph G = (X, A) amounts to looking for the 
number χ iteratively in Cartesian Cartesian graphs,     , q going from |X| to 1. 
 
We work iteratively on q of |X| at 1 while searching  (    ) and when for one i, χ(         ) ≠ |X| andχ ( 
      ) = |X| we have (G)=i. 
Proof: 
The principle is identical to the previous one. We build a graph G
e
 with the complementary of G named   and a 
clique Kq : G
e 
=  + Kq. By iterating on q and looking for the  (G
e
), we find   ( ). 
 If q >   ( ), (Ge)=|X| because all cliques of   have a cardinal lower than the one of clique Kq. 
 Si q ≤   ( ), (Ge)<|X|.  
Thus we can find the number  ( ) with an algorithm to find  . 
In our case, we have an algorithm to find χ, so we reason on the complementary    of Ge, the χ(  ) 
corresponding to  (Ge). 
We obtain the  ( ) because the basis graph is the complementary of G and we have  (G) =  ( ). 
 
 
 
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 : 
   to χ ;    i, 1  i  upper_bound (max degree),   (G+Ki-1)≠|X| and  (G+Ki)=|X|, and χG=i. 
 χ to ;   i, 1  i  |X|, χ (         ) ≠ |X| and χ(       ) = |X|, and (G)=i. 
 
 
 
5. Conclusion 
 
We have presented here how from the chromatic number of any graph find its stability number and vice versa. 
 
These equivalences go through an iterative search on particular graphs consisting of the Cartesian sum of graph 
and a clique. 
 
In another paper, we will show how to transform the chromatic number research into chromatic index research. 
 
 
Xavier Cousin   4/4 
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. 


Somme cartésienne de graphe : pivot entre la recherche du nombre chromatique et celle du nombre de stabilité

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Somme cartésienne de graphe : pivot entre la recherche du nombre chromatique et celle du nombre de stabilité

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