1. Minmax Relations for Cyclically Ordered Graphs
András Sebő,
CNRS, Grenoble

2. -Conj of Gallai (Bessy,Thomassé’64) -Cleaning the notions in it -New results on graphs without cyc. ord.
Algorithms, Polyhedra
max 1Tx : x(S)  1,  S stable, x 0,
min 1Tx : x(C)  1,  dir.cycle C, x 0 integer
Solve them ?
Yes .
But first, put …
… a cyclic order on the vertices

3. G=(V,A) digraph, cover: family F, U F = V
Acyclic iff  order so that every arc is forward
Dilworth : G acyclic, transitive
max stable = min cover by paths(cliques)
Green-Kleitman : G acyclic, transitive
max k-chrom = min  P  P min{ k,|V(P)| }
on covers by paths.

4. Rédei ‘34:  tournament  Hamiltonian path
Camion ‘59:  strong tournament  Ham cycle
Gallai-Roy ‘68:  digraph  (G)-vertex-path.
Bondy ‘76 :  strong ‘’  (G)-vertex-cycle.
? methods for ‘big enough particular cases’ of
stable sets, path partitions, cycle covers,
feedback (arc-)sets, etc. by putting on …
… cyclic orders

5. max k-chrom  min  P  P min{ k, |V(P)|}
Gallai-Milgram (1960): G graph. The vertices of G can be partitioned into at most (G) paths.
Ex-conjecture of Gallai (1962):G strong(ly conn) graph the vertices can be covered by (G) cycles
Thm: Bessy,Thomassé (2003)
Conjecture of Linial :
max k-chrom  min  P  P min{ k, |V(P)|}
on path partitions.
Whose ex-conjecture? In a strong graph with loops:
max k-chrommin  C  C min{k,|V(C)|}c covers
(no loops:max k-chrom min |X| + k | c|: XV,c covers V / X
not partitioned !
(Thm:S.04)

6. structural versions (complementary slackness):
Gallai-Milgram : For each optimal path partition there exists a stable set with one vertex on each path.
BT: G strong => There exists a circuit cover and a stable set with one vertex in each circuit.
Conjecture of Berge: For each path partition minimizing  P  P min{ |V(P)|, k }  a k-colored subgraph where each path meats ’’ ’’ colors.
S.’04 : G strong. There exists a circuit cover and a k-colored subgraph so that each circuit of the cover meets  C  C min{ |V(C)|, k } colors.

7. The winding of a cycle or of a set of cycles:
clockwise
ind(C)=2 C
Bessy,Thomassé: invariance of # through opening !

8. COMPATIBILITY A cyclic order is called compatible, if every arc e
in a cycle, is also in a cycle Ce of winding 1, and the other arcs are forward arcs.
generalizes acyclic: adjacent =>  forward path
Thm (Bessy, Thomassé 2002)  for every digraph
Proof: F (incl-wise) min FAS s.t (|F C|:C cycle) min
G-F acyclic,  compatible order
e  B(ackward arcs) in some shift
F and B are min feedback arc-sets
cycle Ce of G-(B/e): ind(Ce)=1

9. Cyclic stability min { ind(C ) , C cycle cover } clockwise
Bessy,Thomassé: invariance of the index through interchanging nonadjacent consecutive points !
equivalent
S cyclic stable, if stable and interval in equivalent order.
Thm (Bessy, Thomassé 2003) : max cyclic stable=
min { ind(C ) , C cycle cover }

10.  integer primal and dual optimum and in polytime.
x(C)  ind(C)  cycle C, x 0 (BT)
Thm : If the optimum is finite, then
 integer primal and dual optimum and in polytime.
Easy from Bessy-Thomassé through replication.
Easy from mincost flows as well.
But we lost something: the primal has no meaning !
With an additional combinatorial lemma get BT.

11. Get back the lost properties !
Corollary (>~Gallai’s conj): G strong, compatible
=>S stable & C cover such that |S| = ind(C) (|C|)
Proof: uv  E => xu+xv  x(Cuv) ind(Cuv)=1 Q.E.D.
We got back only part of what we have lost: primal
is 0-1, and stable using only
|SC|  1  cycle C with ind(C) =1 .The rest:
Thm: |SC|  i(C)  cycle C <=> S cyclic stable

12. Proof e Algorithm: flow = dual, p(vin) - p(vout) =:xv primal
arcs
backward
vin
vout
if f(e) > lower capacity
lower capacity wv =1
cost = 1 -1 -2
vout
vin 1
If coherent & strong then 0-1
No neg cycle
=>  potential
=: S for which:
From this: primal
Algorithm: flow = dual, p(vin) - p(vout) =:xv primal

13. II. x(S)  1, S cyclic stable, x 0 (antiBT)
Cyclic q-coloring:
G=(V,A)
q=13.28
Place the vertices on a cycle of length q, following an equivalent cyclic order, so that the endpoints of arcs are at ‘distance’  1
min q ?
|C|  ind(C)q
Thm (BT 2003) : min q = max |C| / ind(C)

14. Proof : r := max |C| / ind(C) Define arc-weights: -1 on forward arcs r-1 on backward arcs – |C|+r ind(C) 0  C :No negative cycles potentials … form a coloration + … Q.E.D.
x(C)  ind(C)  cycle C, x  (BT)
x(S)  1  cyclic stable S, x 0 (antiBT)
dual: colorations with cyclic stable sets Thm: Antiblocking pair (with four proofs)

15. Thm 1: x(C) k ind(C)  cycle C, 1  x  0 TDI.
max prim=min |X|+C  C k ind(C) :XV, C covers V\X
=max union of k cyclic stable sets
Thm 2: (BT) has the Integer Decomp Property, i.e.
w  k(BT) int =>w= sum of k integer points in (BT)
Proof:*circ = max |C|/i(C)  *, so = everywhere!
=> * = circ  =  .
w (kBT)=>w/k(BT), that is, max w(C)/ind(C)  k.
By the coloring theorem (after replication) :
w is the sum of k cyclic stable sets. Q.E.D. I

16. Proof: x(C)  k i(C) l  x  u has integer primal, dual, k,l,u
upper=lower capacity=wv
vin
vout
arcs
backward
cost = k
cost = -lv
cost = uv
Etc, Q.E.D
P & IDP & « kP is box TDI »:
Thm: max cyc k-col = min k i(C ) + |not covered|
Proof: kP {x: 0  x  1} = conv {cycl k-col} (IDP) r Formula because of box TDI.
= min{ C  C min{ k ind(C) , |C| }: C cover}

17. III. (blocking) x(C)  i(C)  cycle C, x 0 min 1Tx
cyclic feedback sets: solutions
not consec
Thm : integer primal and dual, and in polytime.
upper capacity w(v) , costs = -1 , …
feedback cyclic feedback
feedback arc cyclic FAS backward arcs 2

18. III. (blocking) x(C)  i(C)  cycle C, x 0 cyclic feedback : min 1Tx
not cyclic
Thm : integer primal and dual, and in polytime.
upper capacity w(v) , costs = -1 , …
feedback cyclic feedback
feedback arc cyclic FAS backward arcs
Attila Bernáth: ‘’ = ‘’ 2

19. Summarizing « Good characterization », and pol algs for
the following variants: choose btw
Antiblocking (containing max cycl stable) blocking (containing min cycl feedback), etc
2. One of the pairs
k=1 or k>1
Vertex or arc version
Arbitrary or transitive

20. The poset of orders (Charbit, S.)
cyclic order 1 ≤ cyclic order 2 (def)
ind 1 (C) ≤ ind 2 (C) for every circuit C.
Exercises: 1. po well-defined on equiv classes
Minimal elements: compatible classes
3. The winding is invariant on any undirected cycle as well – through the operations !

21. Characterizing Equivalence (Charbit, S.)
Problems:
1.* If ind 1 (C) = ind 2 (C) for every undirected
cycle, then order 1 ~ order 2 .
If C is an arbitrary circuit and B(ack arcs)
Then CT  B= |C| - 2 ind.
3. Every C is a linear combination of incidence
vectors of directed circuits .
Thm: If G strongly connected, then order 1 ~ order 2 iff ind 1 (C) = ind 2 (C) for every cycle C.

22. Application: cyclic colorations
r := max |C| / ind(C) Define arc-weights:
-1 on forward arcs, r-1 on backward arcs – -|C|+r ind(C) 0  C : no negative cycles potentials … form a coloration + … Q.E.D.
≥- (r-1) u v -1
r-1
|(u)| < |(v)|
|(v)| < |(u)|
|(v)| =p(v) r + q(v)
uv arc: |p(u)-p(v)| ≤ 1
replace p by q !
Fact: {uv arc: |p(u)-p(v)| = 1} = reversed arcs, cut