THE CACCETTA-H¨ AGGKVIST CONJECTURE Adrian Bondy
What is a beautiful conjecture? The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. G.H. Hardy
Some criteria : ⊲ Simplicity : short, easily understandable statement relating basic concepts. ⊲ Element of Surprise : links together seemingly disparate concepts. ⊲ Generality : valid for a wide variety of objects. ⊲ Centrality : close ties with a number of existing theorems and/or conjectures. ⊲ Longevity : at least twenty years old. ⊲ Fecundity : attempts to prove the conjecture have led to new concepts or new proof techniques.
( d, g ) -cage : smallest d -regular graph of girth g
Lower bound on order of a ( d, g )-cage: order 2( d − 1) r − 2 girth g = 2 r d − 2 order d ( d − 1) r − 2 girth g = 2 r + 1 d − 2 Examples with equality: ⊲ Petersen ⊲ Heawood ⊲ Coxeter-Tutte ⊲ Hoffman-Singleton . . .
We shall consider only oriented graphs: no loops, parallel arcs or directed 2-cycles
Directed ( d, g ) -cage : smallest d -diregular digraph of directed girth g Behzad-Chartrand-Wall Conjecture 1970 The digraph − → d C d ( g − 1)+1 is a directed ( d, g ) -cage Directed (4 , 4) -cage?
Directed ( d, g ) -cage : smallest d -diregular digraph of directed girth g Behzad-Chartrand-Wall Conjecture 1970 The digraph − → d C d ( g − 1)+1 is a directed ( d, g ) -cage Directed (4 , 4) -cage?
Directed ( d, g ) -cage : smallest d -diregular digraph of directed girth g Behzad-Chartrand-Wall Conjecture 1970 The digraph − → d C d ( g − 1)+1 is a directed ( d, g ) -cage Directed (4 , 4) -cage?
Directed ( d, g ) -cage : smallest d -diregular digraph of directed girth g Behzad-Chartrand-Wall Conjecture 1970 The digraph − → d C d ( g − 1)+1 is a directed ( d, g ) -cage Directed (4 , 4) -cage?
Directed ( d, g ) -cage : smallest d -diregular digraph of directed girth g Behzad-Chartrand-Wall Conjecture 1970 The digraph − → d C d ( g − 1)+1 is a directed ( d, g ) -cage Directed (4 , 4) -cage?
COMPOSITIONS Directed (5 , 4) -cage? More generally, if G and H are directed ( d, g ) -cages, then so is their composition G [ H ]
Reformulation: Behzad-Chartrand-Wall Conjecture 1970 Every d -diregular digraph on n vertices has a directed cycle of length at most ⌈ n/d ⌉
VERTEX-TRANSITIVE GRAPHS Hamidoune : In a d -diregular vertex-transitive digraph, there are d directed cycles C 1 , . . . , C d passing through a common vertex, any two meeting only in that vertex: d � | V ( C i ) | ≤ n + d − 1 i =1 � n � So one of these cycles is of length at most d
DISJOINT DIRECTED CYCLES Ho´ ang-Reed Conjecture 1987 In a d -diregular digraph, there are d directed cycles C 1 , . . . , C d such that C j meets ∪ j − 1 i =1 C i in at most one vertex, 1 < j ≤ d . Forest of d Directed Cycles
Mader : Forest of directed cycles not necessarily linear : C d [ C d − 1 ] No linear forest of four directed cycles
Mader : Forest of directed cycles not necessarily linear : C d [ C d − 1 ] No linear forest of four directed cycles
Mader : Forest of directed cycles not necessarily linear : C d [ C d − 1 ] No linear forest of four directed cycles
PRESCRIBED MINIMUM OUTDEGREE Caccetta-H¨ aggkvist Conjecture 1978 Every digraph on n vertices with minimum outdegree d has a directed cycle of length at most ⌈ n/d ⌉ WHAT IS KNOWN? Caccetta and H¨ aggkvist : d = 2 Hamidoune : d = 3 Ho´ ang and Reed : d = 4 , 5 � Shen : d ≤ n/ 2
Chv´ atal and Szemer´ edi : Every digraph on n vertices with minimum outdegree d has a directed cycle of length at most 2 n/d Proof by Induction: ≥ d d v N − ( v ) N + ( v )
≥ d d v N −− ( v ) N − ( v ) N + ( v )
≥ d d v N −− ( v ) N − ( v ) N + ( v )
≥ d d v N −− ( v ) N − ( v ) N + ( v )
≥ d d v N + ( v ) N −− ( v ) N − ( v )
≥ d d v N −− ( v ) N − ( v ) N + ( v )
Chv´ atal and Szemer´ edi : Every digraph on n vertices with minimum outdegree d has a directed cycle of length at most ( n/d ) + 2500 Shen : Every digraph on n vertices with minimum outdegree d has a directed cycle of length at most ( n/d ) + 73 WHAT DOES THIS SAY WHEN d = ⌈ n/ 3 ⌉ ? Every digraph on n vertices with minimum outdegree ⌈ n/ 3 ⌉ has a directed cycle of length at most 76 BUT THE BOUND IN THE CACCETTA-H¨ AGGKVIST CONJECTURE IS 3
Caccetta-H¨ aggkvist Conjecture for triangles Every digraph on n vertices with minimum outdegree ⌈ n/ 3 ⌉ has a directed triangle Caccetta and H¨ aggkvist: Every digraph on n vertices with minimum outdegree √ ⌈ cn ⌉ , where c = 1 2 (3 − 5) , has a directed triangle < (1 − 2 c ) n ≥ cn cn w v N − ( v ) N + ( v ) N ++ ( v ) Assume no directed triangle. Apply induction to subgraph induced by N + ( v ): c 2 − 3 c + 1 > 0 cn ≤ d + ( w ) < c 2 n + (1 − 2 c ) n so
DEGREE BOUNDS FOR A TRIANGLE minimum outdegree ⌈ cn ⌉ : √ aggkvist: c = 1 Caccetta and H¨ 2 (3 − 5) ≈ 0 . 382 √ Shen: c = 3 − 7 ≈ 0 . 3542 minimum indegree and outdegree at least ⌈ cn ⌉ : de Graaf, Seymour and Schrijver: c ≈ . 3487 Shen: c ≈ 0 . 3477
SECOND NEIGHBOURHOODS Seymour’s Second Neighbourhood Conjecture 1990 Every digraph (without directed 2 -cycles) has a vertex with at least as many second neighbours as first neighbours v N + ( v ) N ++ ( v )
The Second Neighbourhood Conjecture implies the triangle case � n � d = 3 of the Behzad-Chartrand-Wall Conjecture ≥ d d d v N + ( v ) N ++ ( v ) N − ( v ) If there is no directed triangle: n ≥ 3 d + 1
Fisher: Second Neighbourhood Conjecture true for tournaments Proof by Havet and Thomass´ e Median order: linear order v 1 , v 2 , . . . , v n maximizing |{ ( v i , v j ) : i < j }| (number of arcs from left to right) Property: for any i ≤ j , vertex v j is dominated by at least half of the vertices v i , v i +1 , . . . , v j − 1 v 1 v i v j v n If not, move v j before v i Claim: | N ++ ( v n ) | ≥ | N + ( v n ) |
v n v n v n
v i v j v n
COUNTING SUBGRAPHS NOTATION D digraph d − ( v ) indegree of v , d outdegree of v , v ∈ V Seven possible types of induced 3-vertex subgraphs: 3 1 2 4 6 7 5
3 1 2 4 6 7 5 x i number of induced subgraphs of type i in D � n � x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = 3 x 2 + 2 x 3 + 2 x 4 + 2 x 5 + 3 x 6 + 3 x 7 = n ( n − 2) d � d − ( v ) � � x 3 + x 6 = 2 v ∈ V x 6 + 3 x 7 = nd 2 x 4 + � d � x 5 + = n x 6 2 Assume no directed triangle: x 7 = 0 Solve in terms of x 6
� n � � d � � d ( v ) � + nd 2 + � x 1 = − n ( n − 2) d + n − x 6 3 2 2 v ∈ V � d � � d ( v ) � − 2 nd 2 − 2 � x 2 = n ( n − 2) d − 2 n + 3 x 6 2 2 v ∈ V � d � x 3 = n − x 6 2 x 4 = nd 2 − x 6 � d ( v ) � � x 5 = − x 6 2 v ∈ V � d � � d ( v ) � − 2 nd 2 − 2 � x 2 + 3 x 3 = n ( n − 2) d + n 2 2 v ∈ V � d � ≤ n ( n − 2) d − 2 nd 2 − n 2 = nd (2 n − 3 − 5 d ) 2 But x 2 ≥ 0 and x 3 ≥ 0, so d ≤ 2 n − 3 5
INDUCED 2-PATHS Thomass´ e’s Conjecture 2006 A digraph on n vertices has at most n 3 15 + 0( n 2 ) induced directed 2 -paths (No condition on degrees or triangles) In our notation: x 4 ≤ n 3 15 + 0( n 2 ) Similar approach to above gives: x 4 ≤ 2 5 x 2 + 1 10 x 3 + x 4 + 1 10 x 5 + 9 2 25 n 3 5 x 7 ≤ Equality: x 1 = 1 150 n 3 , x 2 = 0 , x 3 = 0 , x 4 = 2 25 n 3 x 5 = 0 , x 6 = 2 25 n 3 , x 7 = 0
Roland H¨ aggkvist
Paul Seymour
Vaˇ sek Chv´ atal
Endre Szemer´ edi
Stephan Thomass´ e
References M. Behzad, G. Chartrand and C.E. Wall, On minimal regular digraphs with given girth, Fund. Math. 69 (1970), 227–231. L. Caccetta and R. H¨ aggkvist, On minimal digraphs with given girth, Congressus Numerantium 21 (1978), 181–187. D.C. Fisher, Squaring a tournament: a proof of Dean’s conjecture. J. Graph Theory 23 (1996), 43–48. F. Havet and S. Thomass´ e. Median orders of tournaments: a tool for the second neighborhood problem and Sumner’s conjecture. J. Graph Theory 35 (2000), 244–256. P.D. Seymour, personal communication, 1990. B. Sullivan, A summary of results and problems related to the Caccetta-H¨ aggkvist Conjecture.
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