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The Strong Perfect Graph Theorem

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The Strong Perfect Graph Theorem Annals of Mathematics, 164 (2006), 51–229 The strong perfect graph theorem ∗ ∗ By Maria Chudnovsky, Neil Robertson, Paul Seymour, * ∗∗∗ and Robin Thomas Abstract A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu´ejols and Vuˇskovi´c — that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge’s conjecture cannot have either of these properties). In this paper we prove both of these conjectures. 1. Introduction We begin with definitions of some of our terms which may be nonstandard. All graphs in this paper are finite and simple. The complement G of a graph G has the same vertex set as G, and distinct vertices u, v are adjacent in G just when they are not adjacent in G.Ahole of G is an induced subgraph of G which is a cycle of length at least 4. An antihole of G is an induced subgraph of G whose complement is a hole in G. A graph G is Berge if every hole and antihole of G has even length. A clique in G is a subset X of V (G) such that every two members of X are adjacent. A graph G is perfect if for every induced subgraph H of G, *Supported by ONR grant N00014-01-1-0608, NSF grant DMS-0071096, and AIM. ∗∗ Supported by ONR grants N00014-97-1-0512 and N00014-01-1-0608, and NSF grant DMS-0070912. ∗∗∗ Supported by ONR grant N00014-01-1-0608, NSF grants DMS-9970514 and DMS- 0200595, and AIM. 52 M. CHUDNOVSKY, N. ROBERTSON, P. SEYMOUR, AND R. THOMAS the chromatic number of H equals the size of the largest clique of H. The study of perfect graphs was initiated by Claude Berge, partly motivated by a problem from information theory (finding the “Shannon capacity” of a graph — it lies between the size of the largest clique and the chromatic number, and so for a perfect graph it equals both). In particular, in 1961 Berge [1] proposed two celebrated conjectures about perfect graphs. Since the second implies the first, they were known as the “weak” and “strong” perfect graph conjectures respectively, although both are now theorems: 1.1. The complement of every perfect graph is perfect. 1.2. A graph is perfect if and only if it is Berge. The first was proved by Lov´asz [16] in 1972. The second, the strong perfect graph conjecture, received a great deal of attention over the past 40 years, but remained open until now, and is the main theorem of this paper. Since every perfect graph is Berge, to prove 1.2 it remains to prove the converse. By a minimum imperfect graph we mean a counterexample to 1.2 with as few vertices as possible (in particular, any such graph is Berge and not perfect). Much of the published work on 1.2 falls into two classes: proving that the theorem holds for graphs with some particular graph excluded as an induced subgraph, and investigating the structure of minimum imperfect graphs. For the latter, linear programming methods have been particularly useful; there are rich connections between perfect graphs and linear and integer programming (see [5], [20] for example). But a third approach has been developing in the perfect graph community over a number of years; the attempt to show that every Berge graph either belongs to some well-understood basic class of (perfect) graphs, or admits some feature that a minimum imperfect graph cannot admit. Such a result would therefore prove that no minimum imperfect graph exists, and consequently prove 1.2. Our main result is of this type, and our first goal is to state it. Thus, let us be more precise: we start with two definitions. We say that G is a double split graph if V (G) can be partioned into four sets {a1,...,am}, {b1,...,bm}, {c1,...,cn}, {d1,...,dn} for some m, n ≥ 2, such that: • ai is adjacent to bi for 1 ≤ i ≤ m, and cj is nonadjacent to dj for 1 ≤ j ≤ n. • There are no edges between {ai,bi} and {ai ,bi } for 1 ≤ i<i ≤ m, and all four edges between {cj,dj} and {cj ,dj } for 1 ≤ j<j ≤ n. • There are exactly two edges between {ai,bi} and {cj,dj} for 1 ≤ i ≤ m and 1 ≤ j ≤ n, and these two edges have no common end. (A double split graph is so named because it can be obtained from what is called a “split graph” by doubling each vertex.) The line graph L(G) of a graph G has vertex set the set E(G) of edges of G, and e, f ∈ E(G) are adjacent in THE STRONG PERFECT GRAPH THEOREM 53 L(G) if they share an end in G. Let us say a graph G is basic if either G or G is bipartite or is the line graph of a bipartite graph, or is a double split graph. (Note that if G is a double split graph then so is G.) It is easy to see that all basic graphs are perfect. (For bipartite graphs this is trivial; for line graphs of bipartite graphs it is a theorem of K¨onig [15]; for their complements it follows from Lov´asz’ Theorem 1.1, although originally these were separate theorems of K¨onig; and for double split graphs we leave it to the reader.) Now we turn to the various kinds of “features” that we will prove exist in every Berge graph that is not basic. They are all decompositions of one kind or another, so henceforth we call them that. If X ⊆ V (G) we denote the subgraph of G induced on X by G|X. First, there is a special case of the “2-join” due to Cornu´ejols and Cunningham [13]: a proper 2-join in G is a partition (X1,X2)of V (G) such that there exist disjoint nonempty Ai,Bi ⊆ Xi (i =1, 2) satisfying: • Every vertex of A1 is adjacent to every vertex of A2, and every vertex of B1 is adjacent to every vertex of B2. • There are no other edges between X1 and X2. • For i =1, 2, every component of G|Xi meets both Ai and Bi, and • For i =1, 2, if |Ai| = |Bi| = 1 and G|Xi is a path joining the members of Ai and Bi, then it has odd length ≥ 3. (Thanks to Kristina Vuˇskovi´c for pointing out that we could include the “odd length” condition above with no change to the proof.) If X ⊆ V (G) and v ∈ V (G), we say v is X-complete if v is adjacent to every vertex in X (and consequently v/∈ X), and v is X-anticomplete if v has no neighbours in X.IfX, Y ⊆ V (G) are disjoint, we say X is complete to Y (or the pair (X, Y )iscomplete) if every vertex in X is Y -complete; and being anticomplete to Y is defined similarly. Our second decomposition is a slight variation of the “homogeneous pair” of Chv´atal and Sbihi [7] — a proper homogeneous pair in G is a pair of disjoint nonempty subsets (A, B)ofV (G), such that, if A1,A2 respectively denote the sets of all A-complete vertices and all A-anticomplete vertices in V (G), and B1,B2 are defined similarly, then: • A1 ∪ A2 = B1 ∪ B2 = V (G) \ (A ∪ B) (and in particular, every vertex in A has a neighbour in B and a nonneighbour in B, and vice versa). • The four sets A1 ∩ B1,A1 ∩ B2,A2 ∩ B1,A2 ∩ B2 are all nonempty. A path in G is an induced subgraph of G which is nonnull, connected, not a cycle, and in which every vertex has degree ≤ 2 (this definition is highly nonstandard, and we apologise, but it avoids writing “induced” about 600 times). An antipath is an induced subgraph whose complement is a path. The length of a path is the number of edges in it (and the length of an antipath is the number of edges in its complement). We therefore recognize paths and antipaths of length 0. If P is a path, P ∗ denotes the set of internal vertices 54 M. CHUDNOVSKY, N. ROBERTSON, P. SEYMOUR, AND R. THOMAS of P , called the interior of P ; and similarly for antipaths. Let A, B be disjoint subsets of V (G). We say the pair (A, B)isbalanced if there is no odd path between nonadjacent vertices in B with interior in A, and there is no odd antipath between adjacent vertices in A with interior in B. A set X ⊆ V (G) is connected if G|X is connected (so ∅ is connected); and anticonnected if G|X is connected. The third kind of decomposition used is due to Chv´atal [6] — a skew partition in G is a partition (A, B)ofV (G) such that A is not connected and B is not anticonnected. Despite their elegance, skew partitions pose a difficulty that the other two decompositions do not, for it has not been shown that a minimum imperfect graph cannot admit a skew partition; indeed, this is a well-known open question, first raised by Chv´atal [6], the so-called “skew partition conjecture”.
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