Optimal Linear Inequalities for the Edge and Stability Numbers Jean-Paul Doignon Service de G´ eom´ etrie, Combinatoire et Th´ eorie des Groupes Universit´ e Libre de Bruxelles For a graph G = ( V, E ), denote by n its number of nodes , m its number of links , (thus | V | = n and | E | = m ), α = α ( G ) its stability number (the largest number of two by two nonadjacent nodes in G ). 1
The Petersen Graph has n = 10, m = 15, α = 4. Adding some new link to a graph will leave unchanged or decrease α , deleting a link from a graph will leave unchanged or increase α . On 10 nodes, can we force α ≤ 4 with less than 15 links? 2
Yes: still α = 4. But if we further delete any link: α = 5 > 4. What is the least number of links on 10 nodes that force α ≤ 4? 3
The answer is: m = 8 with a unique solution graph: α = 4 This is the T (10 , 4). It is not connected. Tur´ an Graph Theorem . (Tur´ an, 1941). For n nodes and stability number α , any graph with the minimum possible number of links is a disjoint union of α balanced cliques ( Tur´ an graph T ( n, α )). � n � n � � Balanced means of size or . α α Problem . (Ore, 1962). Same minimizing problem for connected graphs. Apparently open . . . until recently. 4
We consider connected graphs on n nodes, with m links and stability number α . Some examples of valid linear inequalities on m and α (where we consider n as a parameter): m ≥ n − 1 α ≥ 1 α ≤ n − 1 ( n − 2)( n − 3) 2 α + m ≤ + 2 n 2 ( n − 2) α + m ≤ 1 + ( n − 2) n ( n − 2)( n − 3) + n 2 + 1 nα + 2 m ≤ 2 . . . There are infinitely many of them! How can we master this infinite family of linear inequalities? Find a suitable geometric setting! 5
For n fixed, plot in the plane all possible values of ( α, m ). m With n =10: 45 44 42 39 35 30 24 21 17 14 11 9 α 1 2 3 4 5 6 7 8 9 6
Take the convex hull. 45 44 42 39 35 30 24 21 17 The edges of the resulting polygon 14 correspond to 11 optimal linear inequalities . 9 1 2 3 4 5 6 7 8 9 7
Given k graph invariants β 1 , β 2 , . . . , β k , and a class G of graphs, for a fixed number n of nodes, we define in R k P n the polytope of graph invariants β 1 , β 2 , ..., β k ( G ). When the polytope is full-dimensional, it admits a unique description by a minimum system of linear inequalities. These are the optimal linear inequalities . They are finite in number: facet of P n optimal inequality ↔ β 1 , β 2 , ..., β k ( G ). Any linear inequality among the invariants, valid for the class G , is a consequence of the optimal inequalities, in the sense that it is a positive combination of optimal inequalities. Remark. n (the number of nodes) is a parameter. 8
A team consisting of Julie Christophe Sophie Dewez Jean-Paul Doignon Sourour Elloumi Gilles Fasbender Philippe Gr´ egoire David Huygens Martine Labb´ e Hadrien M´ elot Hande Yaman is investigating this polyhedral approach to linear inequalities among graph invariants and has submitted a first paper. 9
Example . ( α, m ) for connected graphs. Here for n = 10: m 45 44 42 39 35 The rightmost edges 30 24 21 The leftmost edges 17 14 11 9 The horizontal edge 9 α 1 2 3 4 5 6 7 8 10
The horizontal edge : x m ≥ n − 1. The rightmost vertices : for a given stability number α , find the largest possible number of links in a connected graph. � n − α � Trivial answer: α ( n − α ) + . 2 The rightmost edges : � n − k � k α + m ≤ + k n , for k = 1, 2, . . . , n − 2. 2 The leftmost vertices : for a given stability number α , find the least possible number of links. Not an easy exercise . . . . . . it is Ore Problem. The rightmost edges : . . . ? Okay, first find the vertices. 11
Ore Problem (1962): What is the least number m of links in connected graphs on n nodes having stability number α ? Definitions . The Tur´ an graph T ( n, α ) is the disjoint union of α balanced cliques � n � (of size ). α The Tur´ an number t ( n, α ) is the number of links of T ( n, α ). Tur´ an Theorem rephrased: Theorem . (Tur´ an, 1941). Any graph G with n nodes and stability number α has at least t ( n, α ) links; if G has t ( n, α ) links, then G is (isomorphic to) the Tur´ an graph T ( n, α ) . Quick conjecture for a “connected Tur´ an Theorem” solving Ore Problem: we need add only α − 1 links to the Tur´ an graph T ( n, α ) to make the graph connected. 12
Example . n = 10, α = 4: m = 8 + 3 Not unique: etc. 13
. . . and even worst: With n = 10, α = 4, other connected graphs which minimize the number of links: m = 11 (= 8 + 3 ?) 14
Theorem . Any connected graph on n nodes with stability number α has at least t ( n, α ) + α − 1 links. This solves Ore’s Problem, and improves on Harant and Schiermeyer (2001). The proof is more involved than the ones for Tur´ an Theorem. Other proofs have been given recently by Gitler and Valencia and by Bougard and Joret who also explain how to generate all critical graphs. 15
Back to the polygon of the invariants α and m for connected graphs: The leftmost points seem to be of the form � � p k = k, t ( n, k ) + k − 1 � n +1 � for k = 1, 2, . . . , . 2 We need to check: (i) Can “convexity” be broken by one of these points? m (i)? (ii)? α 1 2 3 4 5 6 7 8 9 (ii) Can one point lies on the segment joining two other points? (i) No! and (ii) Yes! Investigate how the number of links evolves when k changes in the Tur´ an graph T ( n, k ), more precisely when T ( n, k ) transforms into T ( n, k − 1) or T ( n, k + 1). 16
Lemma . � n � � n � � � p k ∈ p k − 1 , p k +1 ⇐ ⇒ = + 1 . k − 1 k + 1 That is: n n and lie between two consecutive natural numbers. k − 1 k + 1 24 Example . Take n = 24. Mark the values of k : 1 2 3 4 5 6 7 8 9 10 11 12 . 24 /k k = 12 7 5 3 2 Values of k for which p k is a vertex: 1, 2, 3, 4, 5, 6, 8, 12; but not 7, 9, 10, 11. 17
Theorem . The optimal linear inequalities in α and m for connected graphs are m ≥ n − 1; � n − k � k α + m ≤ + k n , for k = 1, 2, . . . , n − 2; 2 � � m − t ( n, k ) − ( k − 1) ≥ t ( n, k ) − t ( n, k − 1) + 1 ( α − k ), � n + 1 � � � � � n n for k = 2, 3, . . . , with � = + 1. 2 k − 1 k + 1 Remark . (e.g. Berge, 1983) �� n � n − k � n �� � � t ( n, k ) = − 1 · . k 2 k 18
Our investigations are computer supported: for small values of n , GraPHedron generates the optimal inequalities, then we formulate conjectures and (hopefully) prove them. Other automated systems exist for investigating invariants of graphs: of Cvetkovi´ c et al. (1980’s); GRAPH of Brigham and Dutton (1980’s); INGRID of Fajtlowicz et al. (late 1980’s); Graffiti of Caporossi and Hansen (1990’s). AutoGraphiX Some systems just provide help to the user, others even build proofs! GraPHedron provides reports for small values of n . It calls geng (Mckay) and ( porta (Christof) or cdd (Fukuda)) as sub- routines. 19
Two other examples investigated by our group, with three invariants: (i) α , m , and ∆ (the maximum degree); (ii) ∆, D (the diameter), and Irr (the irregularity), where � Irr = | deg( i ) − deg( j ) | . { i,j }∈ E We may also change the class G of graphs, for instance taking the class C k of all k -connected graphs. Problem . What is the minimum number of links in a k -connected graph on n nodes, having stability number α ? Solved for k = 2 by Bougard and Joret, open for k ≥ 3. 20
Example . The case of α , m , and ∆ (the maximum degree) for connected graphs on n ≥ 6 vertices. Several optimal inequalities have been obtained, e.g.: m ≥ n − 1 ∆ ≤ n − 1 2 m ≤ n ∆ m ≤ ( n − 3)( n − 2 − α ) + 2∆ ∆ − m − α ≤ 1 − n . . . We still need to work in order to obtain the full list. Here is a Schlegel diagram for the polytope of the invariants α , m , and ∆, when n = 10. 21
Conclusions : a notion of optimal linear inequalities for graph invariants ; a full list of these inequalities in the stability number and the number of edges for the class of connected graphs; a solution to Ore’s Problem; partial lists of optimal inequalities for cases with three invariants. 22
A question . Does anybody know a reference for the following problem? Let X = { ( x, y ) | x, y natural numbers and x · y ≥ n } . What are the vertices of conv X ? Of course, integral points lying on the hyperbola x · y = n are vertices, but there can be other. 23
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