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The chromatic gap and its extremes

• András Sebo, CNRS, G-SCOP,Grenoble
• with
• András Gyárfás MTA SZTAKI, Budapest,
• Nicolas Trotignon, CNRS, LIAFA, Paris

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What is the maximum of the difference
chromatic number max cliquethat can
be reached by graphs on at most n
vertices ?
THE PROBLEM
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?(G) min cover by cliques
chromatic number of complement ?(G) max
stable (independent) set of vertices (no induced
edge) gap (G) ?(G) - ?(G) ? 0 This is the
(integer linear programming) duality gap of a
related linear program
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Perfect, critical, extremal

• Perfect gap 0 ? induced subgraph
• Gap is not necessarily monotonous
• gap-critical G, gap(H)ltgap(G) ? induced H
• Minimal imperfect gap-critical with gap 1.
• t-extremal graph of min order with gap t
• t-extremal ? gap-critical ? factorcritical?

? 3
? 3 gap0
? 2 gap1
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Critique, pas critique ?
G facteur-critique ? C9 ?
C93 ?
G
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s(t) smallest order of a graph of
gap t
gap(n)max gap(G) G has n vertices
gap-extremal ?
gap(i)0 i ?4,
gap-critical !

• ...

gap(5)1, gap(10)2, ???
s(1)5, s(2)10, , s(t) 5t ?
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s(3)15 ? Counterexample

• R13 (3,5)-Ramsey

n13, ?2, ?4, ?7 gap 3
s(3) 13
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Ramsey number, graph

• Ramsey number R(?,?)the smallest integer
  s.t.
• every graph on at least R(?,?) vertices
• has either an ?-clique or an ?-stable set.
• (Ramsey theorem it is finite.)
• Ramsey graph Graph on R(?,?) 1 vertices
  having neither an ?-clique nor an ?-stable set.
• Example R(3,3) 6 (3, 3) Ramsey graph
• R(3,2), R(3,3), 2, 6, 9, 14, 18, 23, 28,
  36, ?
• R(4,4), R(4,5) 18, 25, ?

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EASY FACTS

• Fact 1 If C1 , , Ck the components of G,
• gap(G) gap(C1) gap (Ck)
• G is gap-critical ? all of its components are.
• Fact 2 If G is a graph and Q a clique
• ?(G) ? ?(G-Q) ? ?(G)-1, ?(G) ? ?(G-Q) ? ?(G)-1
• gap(G) 1 ? gap(G-Q) ? gap(G) - 1
• Fact 3 If a t-extremal graph has a k-clique,
• s(t) ? s(t-1) k

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How big should ? be for a big gap ?

• gap ? - ? ? ? n
  / ?
• ? large ? small ? large (Ramsey)
• How does gap ? - ? change if ? varies ?
• Conjecture t-extremal graphs are ?-free
  (?2).
• gap2 (n) max gap of a triangle-free on n
  vertices
• s2 (t) min order of a triangle-free of gap t.
• Theorem s(t) 2t ? (sqrt (t log t ) ) s2
  (t)

gap(n)gap2 (n), s(t)s2 (t) for all n, t
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Inverse Ramsey
?(n) min ?(G) G triangle-free of order
n If R(3, ? ) ? n lt R(3, ? 1 ) ,then ?(n)
? For instance ?(5) 2 , ?(6) 3
Fact If n n1 n2 , then ?(n) ? ?(n1)
?(n2) If n is Ramsey-perfect ?(10)?(5)
?(5) Conjecture This is the only nontriv
example
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A version of the main result

• Theorem ?n / 2? - ?(n) ? gap(n) ? ?n / 2? -
  ?(n)3
• and with the exception of Ramsey numbers
• 1, , 14, there is equality with the lower bound
• The extremal graphs are triangle-free.

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The proof relies on two lucky facts

• (1) gap2 is relatively easy to determine
• (2) Whenever gap gt gap2
• the growth of gap slows down
• while gap2 grows constantly except at Ramsey
• If s (t1) ? s2 (t1), then s (t1) ? s (t)3
• s2 (t1)s2 (t) 2, unless s2 (t)1 or 2 is
  Ramsey

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The gap of triangle-free graphs

• ? ? n / 2
• The more you want, the less you get
• For gap-critical graphs,(if they are connected)
• ? ?n / 2? , so gap ?n / 2? - ?(n)
• 
  
• If not connected, apply to components
• the components are connected, gap-critical

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• If G is gap-critical, ?(G) gt ?(G-v) for all v.
• If G is triangle-free, connected and
• ?(G) gt ?(G-v) for all v, then ?(G) ?n / 2?
• Gallai For triangle-free ? ? n
• Gallai If ? (G-v) ? (G) for every vertex v,
• then G-v has a perfect matching --.
• In particular, n is odd

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Triangle-free, connected contd

• THEN GAP-CRITICAL ?FACTOR-CRITICAL
• Proof of Gallais Lemma
• Equivalent to Tuttes theorem (? Tutte-set)

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The gap of triangle free graphs

• Theorem gap2 (n) ?n / 2? - ?(n) , or 1
• the latter ? n is even Ramsey-perfect.
• Proof At most 2 comp from inequalities about
  Ramsey
• if n n1 n2 n3 , then ?(n) lt ?(n1) ?(n2)
  ?(n3)
• 2 comp if of only if n is even, Ramsey-
  perfect, and the
• components have odd size n1, n2 , ?(n) ?(n1)
  ?(n2)
• For the connected components by Gallai ? ?n /
  2?

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The proof relies on two lucky facts

• (1) gap2 is relatively easy to determine
• (2) Whenever gap gt gap2
• the growth of gap slows down
• while gap2 grows constantly except at Ramsey
• If s (t1) lt s2 (t1), then s (t1)
  ? s (t)3

?n / 2? - ?(n)
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Theorem 0 ? gap(n) - gap2 (n) ? 2and they
are equal everywhere but on small constant size
intervals after Ramsey numbers.

• Corollary All subgraphs of (3, ? ) Ramsey-graphs
  
• of order at least R(3, ?-1) are perfectly
  matchable.

T 1 2 3 4 5 9 10 s(t) 5 10 13
17 20 or 21, 32 or 33 35 R
6, 9, 14, 18, 23,28,
36
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0 ? s2(t) - s (t) ? 10?n / 2?-?(n) ?
gap(n) ? ?n / 2? - ?(n)3
almost always

• NOT YET THE END
• The conjectures ?
• Dramatic corollaries for Ramsey ?
• s(5) 20 or 21 ( s2(5) ) ?