Forbidden Conjectures David Sumner, Professor Emeritus University - PowerPoint PPT Presentation

Forbidden Conjectures David Sumner, Professor Emeritus University of South Carolina

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Conjecture (Matthews, S, 1984). Every 4-connected, claw-free graph is Hamiltonian. Conjecture (Gyárfás, S). For every tree T , there exists a function ( ) . f T such that for every T -free graph G , χ ( G ) ≤ f T ω ( G )

50 Years Earlier

Graphs with no Induced P 4 The earliest work that I am aware of for graphs forbidding paths on four vertices is due to Dave Foulis and Charlie Randall at the University of Massachusetts in their work on Empirical Logic in the early to mid 1960’s. They defined a graph to be a Dacey Graph if whenever two vertices dominate a maximal clique, then they must be adjacent. So, in particular is not a Dacey graph because P 4 K v u

Graphs with no Induced P 4 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. The most common terms for these graphs today seems to be either cographs or . P 4 - free P 4 is self-complementary.

Graphs with no Induced P 4 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. The most common terms for these graphs today seems to be either cographs or . P 4 - free P 4 is self-complementary. The complement of a cograph is again a cograph.

Graphs with no Induced P 4 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 P 4 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 . 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, [ ] . of the connected cograph G , then G = M + G − M

For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining. 
 Graphs with no Induced P 4 A A B B A B Theorem (Wolk 1965). Every cograph is a comparability graph. He did not, of course, use the term cograph or P 4 -free. 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. 
 Graphs with no Induced P 4 Theorem (Wolk 1965). Every cograph is a comparability graph. He did not, of course, use the term cograph or P 4 -free. 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. 4 -free, then χ ( G ) = ω ( G ). Theorem If G is P

For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining. 
 Graphs with no Induced K 1,3 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.

For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining. 
 Graphs with no Induced K 1,3 4 -free, then χ ( G ) = ω ( G ). Theorem If G is P In general, there is no upper bound on the chromatic number of a graph in terms of ω ( G ). Theorem (Erdös) There exist graphs with arbitrarily high girth and chromatic number. Theorem If G is a graph, then χ ( G ) ≤ Δ + 1. 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.

Graphs with no Induced K 1,3 4 -free, then χ ( G ) = ω ( G ). Theorem If G is P Theorem If G is a graph, then χ ( G ) ≤ Δ + 1. Theorem If G is K 1,3 -free, then Δ ( G ) < r (3, ω ( G )) < ω ( G ) 2 An independent set of order 3 would produce a K 1, 3 with center v . v A complete set of order ω would produce a clique on ω + 1 vertices.

Graphs with no Induced K 1,3 4 -free, then χ ( G ) = ω ( G ). Theorem If G is P Theorem If G is a graph, then χ ( G ) ≤ Δ + 1. Theorem If G is K 1,3 -free, then Δ ( G ) < r (3, ω ( G )) < ω ( G ) 2 Theorem If G is K 1,3 -free, then χ ( G ) < ω ( G ) 2 . Theorem If G is P 5 -free and triangle-free, then G is 3-colorable. Theorem If G is P 6 -free, C 6 -free and triangle-free, then G is 3-colorable.

Conjecture (Gyárfás, S). For every tree T , there exists a function ( ) . f T such that for every T -free graph G , χ ( G ) ≤ f T ω ( G ) Theorem . (Gyárfás). The conjecture is true for stars and paths. Theorem . (Kierstead and Penrice). The conjecture is true for trees of radius 2. Theorem . (Kierstead and Zhu). Proved the conjecture for a special class of radius-three trees. Theorem . (Scott). If T is any tree, then there exists a function f T such that every graph that does not contain any subdivision of T as an induced subgraph satisfies χ ( G ) ≤ f T ( ω ( G )).

Conjecture (Gyárfás, S). For every tree T , there exists a function ( ) . f T such that for every T -free graph G , χ ( G ) ≤ f T 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. 
 Graphs with no Induced K 1,3 Theorem . Every even-order, connected, claw-free graph has a 1-factor. This can be proved directly or as a corollary to even stronger results. Theorem . If G is an connected claw-free graph, then it is possible 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. 
 Graphs with no Induced K 1,3 Theorem . Every even-order, connected, claw-free graph has a 1-factor. A referee report from hell! After a year in review... More generally, Theorem . Every even-order, k -connected, claw-free graph with no induced K 1, k + 1 has a 1-factor.

For$any$cograph$G,$at$most$one$of$G$and$$$is$point$determining. 
 Graphs with no Induced K 1,3 Theorem . Every even-order, connected, claw-free graph has a 1-factor. K 1,3 is the only connected graph on four vertices with no 1-factor. Theorem . If G is connected of even order and every induced connected subgraph of order 4 has a 1-factor, then so does G . More generally, Theorem . For any integer k > 1, if G is connected of even order and every induced connected subgraph of order 2 k has a 1-factor, then so does G .

Graphs with no Induced K 1,3 The Oliver Twist Syndrome. Maybe there is more true than the result states. Maybe a slight strengthening of the condition will imply a stronger result. 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.

Graphs with no Induced K 1,3 A non-complete graph G is t - tough if for every separating set S of vertices, S t ≤ Clearly, every Hamiltonian graph is 1-tough. ( ) κ G − S 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 K 1,3 Conjecture (Chvátal ). There exists t > 2, such that if G is t -tough, then G is Hamiltonian. In general, κ ≥ 2 τ for any graph. For claw-free graphs, equality holds. Theorem . (Matthews, S). If G is a claw-free graph, then κ ( G ) = 2 τ ( G ). 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.

Conjecture (Matthews, S, 1984). Every 4-connected, claw-free graph is Hamiltonian. Conjecture (Thomassen, 1985). Every 4-connected line 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. Theorem (Zhan). Every 7-connected, line graph is Hamiltonian. Theorem (Zhan). ) Every 7-connected, claw-free graph is Hamiltonian.

Conjecture (Matthews, S, 1984). Every 4-connected, claw-free graph is Hamiltonian. Conjecture (Thomassen, 1985). Every 4-connected line graph is Hamiltonian. Theorem ( , 1997). The two conjectures are equivalent. There are quite a few other equivalent versions of the conjecture. 
 Including Conjecture ( , Saburov, Vana, 2011). Every 4-connected, claw-free graph is 1-Hamiltonian-connected. ( ) . Every 5-connected, claw-free Theorem Kaiser, Vrána, Rajacek, 2014 graph with minimum degree at least 6 is 1-Hamiltonian-connected.