BEAUTIFUL CONJECTURES IN GRAPH THEORYwkloster/4930/beautiful_conjectures.pdf · What is a beautiful...
Transcript of BEAUTIFUL CONJECTURES IN GRAPH THEORYwkloster/4930/beautiful_conjectures.pdf · What is a beautiful...
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BEAUTIFUL CONJECTURES
IN
GRAPH THEORY
Adrian Bondy
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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
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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.
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Reconstruction ConjectureP.J. Kelly and S.M. Ulam 1942
Every simple graph on at least three vertices isreconstructible from its vertex-deleted subgraphs
STANISLAW ULAM
Simple Surprising General Central Old Fertile
∗∗ ∗ ∗ ∗ ∗ ∗ ∗
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Edge Reconstruction ConjectureF. Harary 1964
Every simple graph on at least four edges isreconstructible from its edge-deleted subgraphs
FRANK HARARY
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗
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MAIN FACTS
Reconstruction Conjecture
False for digraphs. There exist infinitefamilies of nonreconstructible tournaments.
P.J. Stockmeyer 1977
Edge Reconstruction Conjecture
True for graphs on n vertices and morethan n log2 n edges.
L. Lovasz 1972, V. Muller 1977
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Path DecompositionsT. Gallai 1968
Every connected simple graph on n vertices canbe decomposed into at most 1
2(n + 1) paths
TIBOR GALLAI
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗
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Circuit DecompositionsG. Hajos 1968
Every simple even graph on n vertices can bedecomposed into at most 1
2(n − 1) circuits
Gyorgy Hajos
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗
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Hamilton DecompositionsP.J. Kelly 1968
Every regular tournament can be decomposedinto directed Hamilton circuits.
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗
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MAIN FACTS
Gallai’s Conjecture
True for graphs in which all degrees are odd.
L. Lovasz 1968
Hajos’ Conjecture
True for planar graphs and for graphs withmaximum degree four.
J. Tao 1984,
Kelly’s Conjecture
Claimed true for very large tournaments.
R. Haggkvist (unpublished)
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Circuit Double Cover ConjectureP.D. Seymour 1979
Every graph without cut edges has a doublecovering by circuits.
Paul Seymour
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
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Small Circuit Double CoverConjecture
JAB 1990
Every simple graph on n vertices without cutedges has a double covering by at most n − 1
circuits.
JAB
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗
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Cycle Double Cover ConjectureM. Preissmann 1981
Every graph without cut edges has a doublecovering by at most five even subgraphs
Myriam Preissmann
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗
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Matching Double Cover ConjectureR.D. Fulkerson 1971
Every cubic graph without cut edges has a doublecovering by six perfect matchings
REFORMULATION:
Cycle Quadruple Cover ConjectureF. Jaeger 1985
Every graph without cut edges has a quadruplecovering by six even subgraphs
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗
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MAIN FACTS
Circuit Double Cover Conjecture
If false, a minimal counterexample musthave girth at least ten.
L. Goddyn 1988
Small Circuit Double Cover
Conjecture
True for graphs in which some vertex isadjacent to every other vertex.
H. Li 1990
Cycle Double Cover Conjecture
True for 4-edge-connected graphs.
P.A. Kilpatrick 1975, F. Jaeger 1976
True for various classes of snarks.
U. Celmins 1984
Cycle Quadruple Cover Conjecture
Every graph without cut edges has aquadruple covering by seven even subgraphs.
J.C. Bermond, B. Jackson and F. Jaeger 1983
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Five-Flow ConjectureW.T. Tutte 1954
Every graph without cut edges has a 5-flow
Bill Tutte
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
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Three-Flow ConjectureW.T. Tutte 1954
Every 4-edge-connected graph has a 3-flow
Bill Tutte
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗ ∗ ∗ ∗
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WEAKER CONJECTURE:
Weak Three-Flow ConjectureF. Jaeger, 1976
There exists an integer k such that everyk-edge-connected graph has a 3-flow
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MAIN FACTS
Five-Flow Conjecture
Every graph without cut edges has a 6-flow.
P.D. Seymour 1981
Three-Flow Conjecture
Every 4-edge-connected graph has a 4-flow.
F. Jaeger 1976
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Directed CagesM. Behzad, G. Chartrand and C.E. Wall 1970
Every d-diregular digraph on n vertices has adirected circuit of length at most dn/de
Extremal graph for d = dn/3e
(directed triangle)
Simple Surprising General Central Old Fertile
∗∗ ∗ ∗ ∗∗ ∗∗
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Second NeighbourhoodsP.D. Seymour 1990
Every digraph without 2-circuits has a vertexwith at least as many second neighbours as first
neighbours
Paul Seymour
Simple Surprising General Central Old Fertile
∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗
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The Second Neighbourhood Conjectureimplies the case
d =⌈n
3
⌉
of the Directed Cages Conjecture:
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MAIN FACTS
Behzad-Chartrand-Wall Conjecture
Every d-diregular digraph on n vertices hasa directed circuit of length at mostn/d + 2500.
V. Chvatal and E. Szemeredi 1983
True for d ≤ 5.
C. Hoang and B.A. Reed 1987
Every cn-diregular digraph on n verticeswith c ≥ .34615 has a directed triangle.
M. de Graaf 2004
Second Neighbourhood Conjecture
True for tournaments.
J. Fisher 1996, F.Havet and S. Thomasse 2000
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Chords of Longest CircuitsC. Thomassen 1976
Every longest circuit in a 3-connected graph hasa chord
Carsten Thomassen
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗∗ ∗ ∗
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Smith’s ConjectureS. Smith 1984
In a k-connected graph, where k ≥ 2, any twolongest circuits have at least k vertices in
common
Scott Smith
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
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Hamilton Circuits in Line GraphsC. Thomassen 1986
Every 4-connected line graph is hamiltonian
Carsten Thomassen
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
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Hamilton Circuits in Claw-FreeGraphs
M. Matthews and D. Sumner 1984
Every 4-connected claw-free graph is hamiltonian
Simple Surprising General Central Old Prolific
∗ ∗ ∗∗ ∗
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MAIN FACTS
Thomassen’s Chord Conjecture
True for bipartite graphs.
C. Thomassen 1997
Scott Smith’s Conjecture
True for k ≤ 6.
M. Grotschel 1984
Thomassen’s Line Graph Conjecture
Line graphs of 4-edge-connected graphs arehamiltonian.
C. Thomassen 1986
Every 7-connected line graph is hamiltonian.
S.M. Zhan 1991
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Hamilton Circuits in RegularGraphs
J. Sheehan 1975
Every simple 4-regular graph with a Hamiltoncircuit has a second Hamilton circuit
John Sheehan
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗
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AN INTERESTING GRAPH
Used by Fleischner to construct a 4-regularmultigraph with exactly one Hamilton circuit.
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Finding a Second Hamilton CircuitM. Chrobak and S. Poljak 1988
Given a Hamilton circuit in a 3-regular graph,find (in polynomial time) a second Hamilton
circuit
Marek Chrobak and Svatopluk Poljak
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗ ∗ ∗ ∗
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Hamilton Circuits in 4-ConnectedGraphs
H. Fleischner 2004
Every 4-connected graph with a Hamilton circuithas a second Hamilton circuit
Herbert Fleischner
Simple Surprising General Central Old Fertile
∗ ∗ ∗ ∗∗ ∗ ∗
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MAIN FACTS
Sheehan’s Conjecture
Every simple 300-regular graph with aHamilton circuit has a second Hamiltoncircuit.
C. Thomassen 1998
There exist simple uniquely hamiltoniangraphs of minimum degree four.
H. Fleischner 2004
Fleischner’s Conjecture
True for planar graphs.
W.T. Tutte 1956
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What is a beautiful theorem?
Mathematics, rightly viewed, possesses not only truth, but
supreme beauty – a beauty cold and austere, like that of
sculpture.
Bertrand Russell
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.
. Fecundity: has inspired interesting extensions and/or
generalizations.
. Correctness: a beautiful theorem should be true!
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What is a beautiful proof?
. . . an elegant proof is a proof which would not normally come
to mind, like an elegant chess problem: the first move should be
paradoxical . . .
Claude Berge
Claude Berge
Some criteria:
. Elegance: combination of simplicity and surprise.
. Ingenuity: inspired use of standard techniques.
. Originality: introduction of new proof techniques.
. Fecundity: inspires new proof techniques or new proofs of
existing theorems.
. Correctness: a beautiful proof should be correct!
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Most Beautiful ConjectureJ.A.B.
Dominic will continue to prove and conjecturefor many years to come
HAPPY BIRTHDAY, DOMINIC!
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http://www.genealogy.math.ndsu.nodak.edu