DUCHET (Pierre), JANAQI (Stefan), LESCURE (Françoise]., MAAMOUN
(Molaz)., MEYNIEL (Henry), On contractions of digraphs onto K3*, Mat. Bohemica ,
to appear [2002-].
CONTRACTION OF DIGRAPHS ONTO K .
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S. Janaqi
LSD 2-IMAG, CNRS, Université Joseph Fourier, Grenoble, France.
P. Duchet, F. Lescure, M. Maamoun and H. Meyniel
Equipe Combinatoire, CNRS, Université Paris VI, Paris, France.
Abstract :
Extending the notion of a "minor" to directed graphs, we
digraph with vertices, n≥3, and m arcs,
m 3n-3, has
prove that any

3
K as a minor.
§1. Introduction.
We extend the notion of a "minor" to directed graphs. In the non-oriented case
the notion of contraction is well known (see [6] for instance). In this paper Hadwiger
gave the following conjecture :
If  (G)  p then G is contractible onto K p .
Dirac [3] proved this conjecture for
p  4 . Wagner [10] showed that the case p=5
would be a consequence of the 4-color theorem ; Robertson, Seymour and Thomas [8]
proved the case p=6.
For a study of relationships between the existence of some minor of G and a
generalisation of the notion of colouring of digraphs, the interested reader is referred to
[7], where a directed version of Hadwiger’s conjecture is proposed. A sufficient

condition for contractibility onto K3 was given by Duchet and Kaneti [4] in terms of
minimum in- and out- degrees.
We prove here the following :
Theorem 1 : Let G be a digraph with n  3 and m arcs, with

m  3n  3 , then G has K3 as a minor and the bound is the
best
possible for each value of n.
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Duchet and Kaneti [5] gave a long proof of the folowing result : a digraph with n
vertices and at least 5n-8 arcs has K4 as a minor. Theorem 1 might be seen as a
consequence of this result. What we give here is an independent, simpler and shorter
proof.
Most of definitions and notations are standard (see [1]). A digraph G will be
denoted by G=(V, E) where V is the set of its vertices and E is the set of its arcs. We say
that there is a symmetrical arc between u and v if (u,v) and (v,u) both exist. In the case
when there is either (u,v) or (v,u) we say there is a single arc between u and v. We say
that edge uv exists whenever at least one of the arcs (u,v) or (v,u) is present. All the
graphs considered here are finite, connected, without loops or parallel arcs.
The set of out-neighbours (resp. in-neighbours) of a vertex u in the graph G is
denoted by  G (u) (resp.  G (u) ), and we set  G (u)   G (u)   G (u) . We denote by
dG (u) (resp. dG (u) ) the out-degree (resp. in-degree) of a vertex u in the graph G and
dG (u)  dG (u)  dG (u) . We denote by G-x the graph induced by V-x.
We denote by G/e the digraph obtained by identifying the endpoints of an arc
e=(u,v) and by deleting the loops and the parallel arcs created.
A graph G’ obtained from G by a sequence (may be empty) of contractions or
deletions of arcs or deletion of vertices is said to be a minor of G and we note this by
G  G' . This relation is obviously transitive.
§2. Proof of the Theorem :
We give a proof by induction on n+m. The property is obviously true for n=3.
Let G be a graph with at least 4 vertices.
If G has a vertex u with dG (u)  3 simply consider G’=G-u. G’ satisfies the induction
hypothesis, so G'  K3 and since G  G' we have G  K3 . So we may suppose
dG (u)  4 for each vertex u.
If for each vertex u we have dG (u)  6 then G has at least 3n arcs and we can conclude
by the induction hypothesis by deleting any arc.
In the remaining cases we have dG (u)  4 or 5 for some vertex u. We may assume the
following statement holds :
(1) by contracting any arc incident to u we must delete at least 4 arcs,
otherwise we are done by the induction hypothesis.


We may assume that dG (u)  dG (u) The problem could be solved in the same way in
the opposite case.
Suppose first that dG (u)  4 for some vertex u. We can distinguish the following cases :
(i)  G (u)  2 . Let  G (u)  {x,y}. We must have the arcs (x,y) and (y,x) by (1).
(ii)  G (u)  3 . Let  G (u)  {x,y,z} and assume that there is a symmetrical arc between
u and x. By (1) we must have a symmetrical arc between x and z and also the edges uy
and zy. Then contracting either uy or zy we obtain a K3 .
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(iii)  G (u)  4 . In this case we must have at least one arc between any pair of vertices
in  G (u) . If  G (u)   G (u) then  G (u) induces a K4 . If  G (u)  {x,y} and
 G (u)  {z,v} then by identifying x with z and y with v we obtain K3 . Now, if
 G (u)  {x,y,z} and  G (u)  {v} then  G (u) induces a K3 and case dG (u)  4 is done.
Suppose now dG (u)  5 for some vertex u. We may distinguish the following cases :
(i)  G (u)  3 . Let  G (u)  {x,y,z} and let (u,x) be the single arc. By (1) we must have a
symetrical arc between x and y (or x and z) and the edge xz (or xy). So by identifying x
with z (or x with y) we obtain a K3 .
(ii)  G (u)  4 . If  G (u)  {x,y,z,v} and  G (u)  {x} then the subgraph G1 induced by
u   G (u) contains at least 14 arcs and 14 > 3n-3 for n=5. Therefore we can delete one
of these arcs and the induction hypothesis is satisfied for G1 .
Now let us assume that  G (u)  {x,y,z} and G (u)  {x,v}.
Assume that there is at least one symmetrical arc between x and any of the vertices y, z,
v. By symmetry we can choose it to be y. By (1) there must be at least one of vy or vz
and at least one of yz and yv. By contraction of all these arcs we obtain a K3 .
In the other case when there is only one single arc between x and and any of the
vertices y, z, v, by (1) there must be a symmetrical arc between y and z and also the
edge zv. By contracting xy and zv we obtain a K3 .
(iii)  G (u)  5 . If  G (u)  4 or 5 then the subgraph G1 induced by u   G (u)
contains at least 16 arcs. As 16 > 3n-3 for n=6, we are done by the induction hypothesis.


So assume that  G (u)  {x,y,z} and  G (u)  {v,w}. If one of the arcs (a,b), with a in
 G (u) and b in  G (u) , is missing, for instance (w,x), then there must exist (x,y), (x,z),
(v,x), (w,y), (w,z) and (v,w). By identifying y with w and x, z, v we obtain a K3 . If no
such an arc is missing then by contracting (w,x) and (v,y) we obtain a K3 and the case
dG (u)  5 is done.
To see that this bound is the best possible, consider the digraph on n vertices
x1,...., xi ,.....,x n with a symmetrical arc between x1 and xi for i=2,..., n and also the arcs
(xi , x i1 ) for i=2,...,n-1.
Please note that the intuitive conjecture :
"If a digraph G has at least h vertices and at least

(2h-3)-h(h-2) arcs, then G has Kh as a minor"
is false for large values of h. This follows from a work of Bollobas, Catlin, Erdös [2] by
taking a random undirected graph large enough and then replace each edge by two
directed opposite arcs.
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It might be interesting to have a structural characterization of digraphs not
contractible into K3 . A possible step in this direction (as suggested by P. Duchet) would
be to characterise minor-minimal undirected graphs G for which every orientation G' of
G has K3 as a minor.
Concerning contractability of undirected graphs into complete graphs Kostochka
and then Thomason (see [9]) gave a good deterministic lower bound depending on the
edge density. It would be interesting to have the corresponding result for directed
graphs. An analoguous good bound for the directed case does not follow in a simple
manner from Thomason's result (by contracting into a large tournament and then
contracting into K p , we need more than the expected number of edges).
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[4] P. Duchet, V. Kaneti, Sur la contraction des graphes orientés en K3 , Discrete
Mathematics, Vol. 65, No. 1 (1987), p. 95-98.

[5] P. Duchet, V. Kaneti, Sur la contractibilité d’un graphe orienté en K4 , Discrete
Mathematics, 130 (1994), 12 pp., in print.
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