Forbidden Conjectures David Sumner, Professor Emeritus University - - PowerPoint PPT Presentation
Forbidden Conjectures David Sumner, Professor Emeritus University of South Carolina Graph Theory app for iPad coming soon to the Apple app store. ! Email david@dpsumner.com for more information or to request to be a beta-tester. See more
David Sumner, Professor Emeritus University of South Carolina
See more details at http://www.dpsumner.com Graph Theory app for iPad coming soon to the Apple app store.
!
Email david@dpsumner.com for more information or to request to be a beta-tester.
Conjecture (Matthews, S, 1984). Every 4-connected, claw-free graph is Hamiltonian.
Conjecture (Gyárfás, S). For every tree T, there exists a function fT such that for every T-free graph G, χ(G) ≤ fT ω(G)
( ).
The earliest work that I am aware of for graphs forbidding paths
University of Massachusetts in their work on Empirical Logic in the early to mid 1960’s.
Graphs with no Induced P
4
So, in particular is not a Dacey graph because
P
4
v u K
They defined a graph to be a Dacey Graph if whenever two vertices dominate a maximal clique, then they must be adjacent.
The most common terms for these graphs today seems to be either cographs or .
P
4-free
Foulis - Randall defined a graph G to be hereditary Dacey if every induced subgraph of G is a Dacey Graph. It turns out that the hereditary Dacey graphs are precisely those graphs that have no induced path on four vertices.
Graphs with no Induced P
4
P
4 is self-complementary.
The most common terms for these graphs today seems to be either cographs or .
P
4-free
Foulis - Randall defined a graph G to be hereditary Dacey if every induced subgraph of G is a Dacey Graph. It turns out that the hereditary Dacey graphs are precisely those graphs that have no induced path on four vertices.
Graphs with no Induced P
4
P
4 is self-complementary.
The complement of a cograph is again a cograph.
Theorem (Foulis - Randall). If G is a cograph, then there exist cographs A and B such that (i). If G is connected, then G = A + B. (ii). If G is not connected, then G = A∪ B.
A B A B
G is connected G is not connected G is the join of A and B. G is the union of A and B.
Graphs with no Induced P4
A B A B
G is connected G is not connected G is the join of A and B. G is the union of A and B.
Graphs with no Induced P4
Theorem (S, 1971). If S is a maximal independent set of the connected cograph G, then N(S) ≠ ∅ and G = N(S)+ G − N(S)
[ ].
Theorem (S, 1971). If M is a minimal separating set of vertices,
[ ].
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
Graphs with no Induced P4
He did not, of course, use the term cograph or P
4-free.
Theorem (Wolk 1965). Every cograph is a comparability graph.
A B A B A B
Orient each of A and B transitively. Orient edges between A and B from A towards B.
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
Theorem (Seinsche 1974). Every cograph is perfect. Arditti and DeWerra pointed out in a note in JCT(B) 1976 that Seinsche’s result was really a special case of Wolk’s.
Graphs with no Induced P4
He did not, of course, use the term cograph or P
4-free.
Theorem If G is P
4-free, then χ(G) = ω(G).
Theorem (Wolk 1965). Every cograph is a comparability graph.
Graphs with no Induced K1,3
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
The other tree on four vertices is the claw and the graphs that do not contain an induced claw are even more interesting than the cographs. The first mention of the claw-free graphs that I saw was in Beineke’s forbidden subgraph characterization of line graphs.
Graphs with no Induced K1,3
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
Theorem If G is P
4-free, then χ(G) = ω(G).
Theorem If G is a graph, then χ(G) ≤ Δ +1. In general, there is no upper bound on the chromatic number
Theorem (Erdös) There exist graphs with arbitrarily high girth and chromatic number. Ramsey Number: For positive integers n,m every graph with more than r(n,m) vertices has an independent set on n vertices or a complete subgraph on m vertices.
An independent set of order 3 would produce a K1, 3 with center v.
Graphs with no Induced K1,3
Theorem If G is P
4-free, then χ(G) = ω(G).
Theorem If G is K1,3-free, then Δ(G) < r(3, ω(G))< ω(G)2 Theorem If G is a graph, then χ(G) ≤ Δ +1.
v
A complete set of order ω would produce a clique on ω +1 vertices.
Theorem If G is P
4-free, then χ(G) = ω(G).
Theorem If G is K1,3-free, then Δ(G) < r(3, ω(G))< ω(G)2 Theorem If G is a graph, then χ(G) ≤ Δ +1. Theorem If G is P
5-free and triangle-free, then G is 3-colorable.
Theorem If G is K1,3-free, then χ(G) < ω(G)2.
Graphs with no Induced K1,3
Theorem If G is P
6-free, C6-free and triangle-free, then G is 3-colorable.
such that every graph that does not contain any subdivision of T as an induced subgraph satisfies χ(G) ≤ fT (ω(G)). Conjecture (Gyárfás, S). For every tree T, there exists a function fT such that for every T-free graph G, χ(G) ≤ fT ω(G)
( ).
class of radius-three trees.
Conjecture (Gyárfás, S). For every tree T, there exists a function fT such that for every T-free graph G, χ(G) ≤ fT ω(G)
( ).
If we don't care about the trees being induced, then much stronger results are true. Theorem (Gyárfás, S). Every coloring of a k-chromatic graph using the labels {1,2,…,k} contains a copy of every labelled tree on {1,2,…,k}.
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
has a 1-factor.
Graphs with no Induced K1,3
This can be proved directly or as a corollary to even stronger results.
to produce a maximum matching for G by sequentially removing adjacent pairs of vertices so that the graph remains connected after each deletion.
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
has a 1-factor.
Graphs with no Induced K1,3
with no induced K1,k+1 has a 1-factor. A referee report from hell! After a year in review...
More generally,
For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining.
has a 1-factor.
Graphs with no Induced K1,3
connected subgraph of order 4 has a 1-factor, then so does G.
and every induced connected subgraph of order 2k has a 1-factor, then so does G.
More generally,
K1,3 is the only connected graph on four vertices with no 1-factor.
Graphs with no Induced K1,3
A graph is locally-connected if the open neighborhood, N(v), of each vertex induces a connected subgraph. Theorem (Oberly, S). If G is a connected, locally-connected, claw-free graph, then G is Hamiltonian. Maybe there is more true than the result states. Maybe a slight strengthening of the condition will imply a stronger result. The Oliver Twist Syndrome.
Graphs with no Induced K1,3
A non-complete graph G is t-tough if for every separating set S of vertices,
t ≤ S κ G − S
( )
Clearly, every Hamiltonian graph is 1-tough. Conjecture (Chvátal ). There exists t > 1, such that if G is t-tough, then G is Hamiltonian. Conjecture (Chvátal ). There exists t > 2, such that if G is t-tough, then G is Hamiltonian.
Graphs with no Induced K1,3
Conjecture (Chvátal ). There exists t > 2, such that if G is t-tough, then G is Hamiltonian.
In general, κ ≥ 2τ for any graph.
Theorem (Tutte). Every 4-connected, planar graph is Hamiltonian. Theorem (Tutte). If G is planar with τ(G) > 32, then G is Hamiltonian.
Moreover, if the 2-tough conjecture were true, then 4-connected would be enough to guarantee claw-free graphs are Hamiltonian.
For claw-free graphs, equality holds.
Conjecture (Matthews, S, 1984). Every 4-connected, claw-free graph is Hamiltonian. Conjecture (Fleischner, 1996). These two conjectures are equivalent. Theorem ( , 1997). The two conjectures are equivalent. He proved this result using a new closure concept. Conjecture (Thomassen, 1985). Every 4-connected line graph is Hamiltonian.
Theorem (Zhan). Every 7-connected, line graph is Hamiltonian. Theorem (Zhan). ) Every 7-connected, claw-free graph is Hamiltonian.
Theorem ( , 1997). The two conjectures are equivalent. Conjecture (Thomassen, 1985). Every 4-connected line graph is Hamiltonian. Conjecture ( , Saburov, Vana, 2011). Every 4-connected, claw-free graph is 1-Hamiltonian-connected. There are quite a few other equivalent versions of the conjecture. Including
Theorem Kaiser, Vrána, Rajacek, 2014
( ). Every 5-connected, claw-free
graph with minimum degree at least 6 is 1-Hamiltonian-connected.
Conjecture (Matthews, S, 1984). Every 4-connected, claw-free graph is Hamiltonian.
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