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Robin Thomas 1 THE STRONG PERFECT GRAPH THEOREM Robin Thomas School of Mathematics Georgia Institute of Technology and American Institute of Mathematics joint work with M. Chudnovsky, Neil Robertson and P. D. Seymour 2 PART I History and relevance of perfect graphs PART II The strong perfect graph conjecture 3 CLAUDE BERGE 1926{2002 4 Graphs have vertices 5 Graphs have vertices and edges. 6 Graphs have vertices and edges. A clique is a set of pairwise adjacent vertices. 7 Graphs have vertices and edges. A clique is a set of pairwise adjacent vertices. !(H) = size of maximum clique of H 8 Graphs have vertices and edges. A clique is a set of pairwise adjacent vertices. !(H) = size of maximum clique of H In a coloring adjacent vertices receive different colors. 9 Graphs have vertices and edges. A clique is a set of pairwise adjacent vertices. !(H) = size of maximum clique of H In a coloring adjacent vertices receive different colors. 9 Graphs have vertices and edges. A clique is a set of pairwise adjacent vertices. !(H) = size of maximum clique of H In a coloring adjacent vertices receive different colors. χ(H) = minimum number of colors needed 9 Graphs have vertices and edges. A clique is a set of pairwise adjacent vertices. !(H) = size of maximum clique of H In a coloring adjacent vertices receive different colors. χ(H) = minimum number of colors needed Clearly χ(H) !(H). ≥ 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). ≥ 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). A graph G is perfect if ≥ χ(H) = !(H) for every induced subgraph H. 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). A graph G is perfect if ≥ χ(H) = !(H) for every induced subgraph H. DEFINITION A hole is a cycle of length at least four; its complement is an antihole.A hole/antihole in G is an induced subgraph that is a hole/antihole. 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). A graph G is perfect if ≥ χ(H) = !(H) for every induced subgraph H. DEFINITION A hole is a cycle of length at least four; its complement is an antihole.A hole/antihole in G is an induced subgraph that is a hole/antihole. GRAPHS THAT ARE NOT PERFECT 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). A graph G is perfect if ≥ χ(H) = !(H) for every induced subgraph H. DEFINITION A hole is a cycle of length at least four; its complement is an antihole.A hole/antihole in G is an induced subgraph that is a hole/antihole. GRAPHS THAT ARE NOT PERFECT Odd holes 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). A graph G is perfect if ≥ χ(H) = !(H) for every induced subgraph H. DEFINITION A hole is a cycle of length at least four; its complement is an antihole.A hole/antihole in G is an induced subgraph that is a hole/antihole. GRAPHS THAT ARE NOT PERFECT Odd holes Odd antiholes 10 χ(H) = minimum number of colors needed !(H) = maximum size of a clique Clearly χ(H) !(H). A graph G is perfect if ≥ χ(H) = !(H) for every induced subgraph H. DEFINITION A hole is a cycle of length at least four; its complement is an antihole.A hole/antihole in G is an induced subgraph that is a hole/antihole. GRAPHS THAT ARE NOT PERFECT Odd holes Odd antiholes Graphs that have an odd hole or odd antihole 11 EXAMPLES OF PERFECT GRAPHS 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs (K¨onig1916) 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs (K¨onig1916) their complements 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs (K¨onig1916) their complements (K¨onig1931) 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs (K¨onig1916) their complements (K¨onig1931) Etc::: There are 96 known classes 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs (K¨onig1916) their complements (K¨onig1931) Etc::: There are 96 known classes THE PERFECT GRAPH THEOREM (Lov´asz1972) A graph is perfect its complement is perfect. , 11 EXAMPLES OF PERFECT GRAPHS Bipartite graphs (! = 2 = χ) their complements (K¨onig,Egerv´ary1931) Line graphs of bipartite graphs (K¨onig1916) their complements (K¨onig1931) Etc::: There are 96 known classes THE PERFECT GRAPH THEOREM (Lov´asz1972) A graph is perfect its complement is perfect. , THE STRONG PERFECT GRAPH CONJECTURE (SPGC) (Berge 1960) A graph is perfect it has no odd hole and no odd , antihole (\Berge graph") 12 BERGE'S MOTIVATION 12 BERGE'S MOTIVATION Consider a discrete memoryless channel. Elements of a finite alphabet Σ are transmitted, some pairs of elements may be confused. 12 BERGE'S MOTIVATION Consider a discrete memoryless channel. Elements of a finite alphabet Σ are transmitted, some pairs of elements may be confused. EXAMPLE Σ = a; b; c; d; e , ab; bc; cd; de; ea may be f g confused. 12 BERGE'S MOTIVATION Consider a discrete memoryless channel. Elements of a finite alphabet Σ are transmitted, some pairs of elements may be confused. EXAMPLE Σ = a; b; c; d; e , ab; bc; cd; de; ea may be f g confused. So a; c may be sent without confusion 12 BERGE'S MOTIVATION Consider a discrete memoryless channel. Elements of a finite alphabet Σ are transmitted, some pairs of elements may be confused. EXAMPLE Σ = a; b; c; d; e , ab; bc; cd; de; ea may be f g confused. So a; c may be sent without confusion 2n n-symbol error-free messages ) 12 BERGE'S MOTIVATION Consider a discrete memoryless channel. Elements of a finite alphabet Σ are transmitted, some pairs of elements may be confused. EXAMPLE Σ = a; b; c; d; e , ab; bc; cd; de; ea may be f g confused. So a; c may be sent without confusion 2n n-symbol error-free messages ) But ab; bd; ca; dc; ee are pairwise unconfoundable 12 BERGE'S MOTIVATION Consider a discrete memoryless channel. Elements of a finite alphabet Σ are transmitted, some pairs of elements may be confused. EXAMPLE Σ = a; b; c; d; e , ab; bc; cd; de; ea may be f g confused. So a; c may be sent without confusion 2n n-symbol error-free messages ) But ab; bd; ca; dc; ee are pairwise unconfoundable 1 5n=2 = 2(2 log 5)n n-symbol error-free messages ) 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n We have !n(G) !(Gn) χ(Gn) χn(G) ≤ ≤ ≤ 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n We have !n(G) !(Gn) χ(Gn) χn(G) ≤ ≤ ≤ and so if !(G) = χ(G), then they are equal to C(G). 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n We have !n(G) !(Gn) χ(Gn) χn(G) ≤ ≤ ≤ and so if !(G) = χ(G), then they are equal to C(G). 1 Lov´asz proved that C(C5) = 2 log 5 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n We have !n(G) !(Gn) χ(Gn) χn(G) ≤ ≤ ≤ and so if !(G) = χ(G), then they are equal to C(G). 1 Lov´asz proved that C(C5) = 2 log 5, using geometric representations of graphs (theta function). 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n We have !n(G) !(Gn) χ(Gn) χn(G) ≤ ≤ ≤ and so if !(G) = χ(G), then they are equal to C(G). 1 Lov´asz proved that C(C5) = 2 log 5, using geometric representations of graphs (theta function). C(G) of many graphs is not known. 13 Let V (G) = Σ, where a; b adjacent if unconfoundable 1 n Shannon capacity C(G) := limn log !(G ) !1 n We have !n(G) !(Gn) χ(Gn) χn(G) ≤ ≤ ≤ and so if !(G) = χ(G), then they are equal to C(G). 1 Lov´asz proved that C(C5) = 2 log 5, using geometric representations of graphs (theta function). C(G) of many graphs is not known. 14 THE RELEVANCE OF PERFECT GRAPHS Generalizations of classical theorems about graphs • 14 THE RELEVANCE OF PERFECT GRAPHS Generalizations of classical theorems about graphs • Communication theory (Shannon capacity, entropy) • 14 THE RELEVANCE OF PERFECT GRAPHS Generalizations of classical theorems about graphs • Communication theory (Shannon capacity, entropy) • Polyhedral combinatorics • 14 THE RELEVANCE OF PERFECT GRAPHS Generalizations of classical theorems about graphs • Communication theory (Shannon capacity, entropy) • Polyhedral combinatorics • Relation to integrality of polyhedra • 14 THE RELEVANCE OF PERFECT GRAPHS Generalizations of classical theorems about graphs • Communication theory (Shannon capacity, entropy) • Polyhedral combinatorics • Relation to integrality of polyhedra • Geometric algorithms of Gr¨otschel,Lov´asz,Schrijver • 14 THE RELEVANCE OF PERFECT GRAPHS Generalizations of classical theorems about graphs • Communication theory
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