CONTRACTION OF DIGRAPHS ONTO K 3.

Transcription

1 DUCHET (Pierre), JANAQI (Stefan), LESCURE (Françoise], MAAMOUN (Molaz), MEYNIEL (Henry), On contractions of digraphs onto K 3 *, Mat Bohemica, to appear [2002-] CONTRACTION OF DIGRAPHS ONTO K 3 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 prove that any digraph with vertices, n 3, and m arcs, m 3n-3, has K 3 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 χ(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 K 3 was given by Duchet and Kaneti [4] in terms of minimum in- and out- degrees We prove here the following : 1

2 Theorem 1 : Let G be a digraph with n 3 and m arcs, with m 3n 3, then G has K 3 as a minor and the bound is the best possible for each value of n Duchet and Kaneti [5] gave a long proof of the folowing result : a digraph with n vertices and at least 5n-8 arcs has K 4 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 Γ + G (u) (resp Γ G (u)), and we set Γ G (u) = Γ + G (u) Γ G (u) We denote by d + G (u) (resp d G (u)) the out-degree (resp in-degree) of a vertex u in the graph G and d G (u) = d + G (u) + d G (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 d G (u) 3 simply consider G =G-u G satisfies the induction hypothesis, so G' K 3 and since G G' we have G K 3 So we may suppose d G (u) 4 for each vertex u If for each vertex u we have d G (u) 6 then G has at least 3n arcs and we can conclude by the induction hypothesis by deleting any arc 2

3 In the remaining cases we have d G (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 d + G (u) d G (u) The problem could be solved in the same way in the opposite case Suppose first that d G (u) = 4 for some vertex u We can distinguish the following cases : (i) Γ G (u) = 2 Let Γ G (u) = {x, y} We must have the arcs (x,y) and (y,x) by (1) (ii) Γ G (u) = 3 Let Γ 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 K 3 (iii) Γ G (u) = 4 In this case we must have at least one arc between any pair of vertices in Γ G (u) If Γ G (u) = Γ + G (u) then Γ G (u) induces a K 4 If Γ + G (u) = {x, y} and Γ G (u) = {z,v} then by identifying x with z and y with v we obtain K 3 Now, if Γ + G (u) = {x, y,z} and Γ G (u) = {v} then Γ + G (u) induces a K 3 and case d G (u) = 4 is done Suppose now d G (u) = 5 for some vertex u We may distinguish the following cases : (i) Γ G (u) = 3 Let Γ 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 K 3 (ii) Γ G (u) = 4 If Γ + G (u) = {x, y,z,v} and Γ G (u) = {x} then the subgraph G 1 induced by u Γ 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 G 1 Now let us assume that Γ + G (u) = {x, y,z} and Γ 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 K 3 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 3

4 arc between y and z and also the edge zv By contracting xy and zv we obtain a K 3 (iii) Γ G (u) = 5 If Γ + G (u) = 4 or 5 then the subgraph G 1 induced by u Γ G (u) contains at least 16 arcs As 16 > 3n-3 for n=6, we are done by the induction hypothesis So assume that Γ + G (u) = {x, y,z} and Γ G (u) = {v,w} If one of the arcs (a,b), with a in Γ G (u) and b in Γ + 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 K 3 If no such an arc is missing then by contracting (w,x) and (v,y) we obtain a K 3 and the case d G (u) = 5 is done To see that this bound is the best possible, consider the digraph on n vertices x 1,,x i,,x n with a symmetrical arc between x 1 and x i for i=2,, n and also the arcs (x i, 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 K h 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 It might be interesting to have a structural characterization of digraphs not contractible into K 3 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 K 3 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) REFERENCES : [1] C Berge, Graphs and Hypergraphs North Holland Mathematical Library 6, Amsterdam, London, 528 pp,

5 [2] B Bollobas, P A Catlin, P Erdös, Hadwiger s conjecture is true for almost every graph, Europ J Combinatorics, 1, 1980, [3] GA Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J of London Math Soc 27 (1952) [4] P Duchet, V Kaneti, Sur la contraction des graphes orientés en K 3, Discrete Mathematics, Vol 65, No 1 (1987), p [5] P Duchet, V Kaneti, Sur la contractibilité d un graphe orienté en K 4, Discrete Mathematics, 130 (1994), 12 pp, in print [6] H Hadwiger, Uber eine Klassifikation der Streckenkcomplexe, Vierteljschr Naturforsch Ges Zürich 88 (1943) [7] H Jacob, H Meyniel, Extention of Turan s and Brooks theorems and new notions of stability and colouring in digraphs, Annals of Discrete Mathematics 17, 1983, [8] Robertson N, Seymour P, Thomas R, Hadwiger s conjecture for K 6 -free graphs Combinatorica 13, (1993), 3, p [9] A Thomason, An extremal function for contractions of graphs, Math Proc Camb Phil Soc 95 (1984) [10] K Wagner, Bemerkungen zu Hadwigers Vermutung, Math Ann 141 (1960)