Matthews-Sumner Conjecture and Equivalences

Transcription

1 University of Memphis June 21, 2012

2 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F.

3 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F. Forbidden subgraphs have been studied relative to various Hamiltonian type properties

4 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F. Forbidden subgraphs have been studied relative to various Hamiltonian type properties Traceable,

5 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F. Forbidden subgraphs have been studied relative to various Hamiltonian type properties Traceable, Hamiltonian,

6 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F. Forbidden subgraphs have been studied relative to various Hamiltonian type properties Traceable, Hamiltonian, Hamiltonian Connected,

7 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F. Forbidden subgraphs have been studied relative to various Hamiltonian type properties Traceable, Hamiltonian, Hamiltonian Connected, Pancyclic,

8 Forbidden Subgraphs Definition A graph G is H-free if G contains no induced copy of the graph H as a subgraph. More generally, we say G is F-free for some family of connected graphs F, provided G contains no induced subgraph isomorphic to a graph in F. Forbidden subgraphs have been studied relative to various Hamiltonian type properties Traceable, Hamiltonian, Hamiltonian Connected, Pancyclic, Panconnected.

9 The Matthews-Sumner Conjecture, 1984 Conjecture A Every 4-connected claw-free graph is hamiltonian.

10 Thomassen s Conjecture, 1986 Conjecture B Every 4-connected line graph is hamiltonian.

11 Thomassen s Conjecture, 1986 Conjecture B Every 4-connected line graph is hamiltonian. This conjecture was mentioned as early as 1981

12 4-connected is necessary Theorem (Jackson, Wormald) Let G be a 3-connected K 1,r -free graph with n vertices. Then G contains a cycle of length at least n c where c = (log26+2log2(2r 1)) 1.

13 4-connected is necessary Theorem (Jackson, Wormald) Let G be a 3-connected K 1,r -free graph with n vertices. Then G contains a cycle of length at least n c where c = (log26+2log2(2r 1)) 1. There are infinite families of examples obtained by taking the inflations (replacing a vertex by a triangle) of appropriate 3-regular graphs (like the Petersen graph) that imply that the Jackson-Wormald bound is of the correct order of magnitude (but not necessarily with the constant c).

14 EIDMA workshop on Hamiltonicity of 2-tough graphs 1996 Line graphs are claw-free, so the Matthews-Sumner Conjecture implies the Thomassen conjecture.

15 EIDMA workshop on Hamiltonicity of 2-tough graphs 1996 Line graphs are claw-free, so the Matthews-Sumner Conjecture implies the Thomassen conjecture. Herbert Fleischner conjectured that the two conjectures were equivalent.

16 EIDMA workshop on Hamiltonicity of 2-tough graphs 1996 Line graphs are claw-free, so the Matthews-Sumner Conjecture implies the Thomassen conjecture. Herbert Fleischner conjectured that the two conjectures were equivalent. This was verified by Zdenek Ryjáček and the result appeared in 1997.

17 Present Ryjáček Closure Theorem (Ryjáček. 1997) Let G be a claw-free graph. Then

18 Present Ryjáček Closure Theorem (Ryjáček. 1997) Let G be a claw-free graph. Then the closure cl(g) is uniquely determined,

19 Present Ryjáček Closure Theorem (Ryjáček. 1997) Let G be a claw-free graph. Then the closure cl(g) is uniquely determined, the circumference of cl(g) is equal to the circumference of G,

20 Present Ryjáček Closure Theorem (Ryjáček. 1997) Let G be a claw-free graph. Then the closure cl(g) is uniquely determined, the circumference of cl(g) is equal to the circumference of G, cl(g) is hamiltonian if and only if G is hamiltonian,

21 Present Ryjáček Closure Theorem (Ryjáček. 1997) Let G be a claw-free graph. Then the closure cl(g) is uniquely determined, the circumference of cl(g) is equal to the circumference of G, cl(g) is hamiltonian if and only if G is hamiltonian, cl(g) is the line graph of a triangle-free graph.

22 Present Ryjáček Closure Theorem (Ryjáček. 1997) Let G be a claw-free graph. Then the closure cl(g) is uniquely determined, the circumference of cl(g) is equal to the circumference of G, cl(g) is hamiltonian if and only if G is hamiltonian, cl(g) is the line graph of a triangle-free graph. This closure was patterned after the Bondy - Chvátal closure relative to Ore s Theorem.

23 Chvátal s Conjecture, 1973 Toughness Conjecture There is a positive k such that if t(g) > k, then G is Hamiltonian.

24 Chvátal s Conjecture, 1973 Toughness Conjecture There is a positive k such that if t(g) > k, then G is Hamiltonian. The value of k = 3/2 initially proposed.

25 Chvátal s Conjecture, 1973 Toughness Conjecture There is a positive k such that if t(g) > k, then G is Hamiltonian. The value of k = 3/2 initially proposed. For some time the value k = 2 was investigated, which was interesting since t(g) 2 for any 4-connected claw-free graph.

26 Chvátal s Conjecture, 1973 Toughness Conjecture There is a positive k such that if t(g) > k, then G is Hamiltonian. The value of k = 3/2 initially proposed. For some time the value k = 2 was investigated, which was interesting since t(g) 2 for any 4-connected claw-free graph. The Chvátal Conjecture is still open.

27 Present Status Theorem (Zhan, 1991) Every 7-connected line graph is hamiltonian.

28 Present Status Theorem (Zhan, 1991) Every 7-connected line graph is hamiltonian. This was proved independently by Bill Jackson, but not published.

29 Present Status Theorem (Zhan, 1991) Every 7-connected line graph is hamiltonian. This was proved independently by Bill Jackson, but not published. Theorem (Kaiser and Vrana, 2011) Every claw-free, 5-connected graph with minimum degree 6 is hamiltonian connected, and hence hamiltonian.

30 Hamiltonicity in Line Graphs Definition A circuit is a closed walk in a graph that does not repeat edges.

31 Hamiltonicity in Line Graphs Definition A circuit is a closed walk in a graph that does not repeat edges. Definition A dominating circuit is a circuit in which each edge of the graph is either on the circuit, or incident to a vertex on the circuit.

32 Hamiltonicity in Line Graphs Definition A circuit is a closed walk in a graph that does not repeat edges. Definition A dominating circuit is a circuit in which each edge of the graph is either on the circuit, or incident to a vertex on the circuit. Theorem (Harary and Nash-Williams, 1965) Let H be a graph with at least 3 edges. Then L(H) is hamiltonian if and only if H = K 1,r with r 3 or H contains a dominating circuit.

33 Appropriate Edge-Connectivity for 4-connected Line Graph Question What is the counterpart of 4-connectivity in L(H)?

34 Appropriate Edge-Connectivity for 4-connected Line Graph Question What is the counterpart of 4-connectivity in L(H)? Definition A graph H is essentially 4-edge connected if it contains no edge-cut R such that R < 4 and at least two components of H R contain an edge.

35 The Dominating Circuit Conjecture Theorem L(H) is 4-connected if and only if H is essentially 4-edge connected.

36 The Dominating Circuit Conjecture Theorem L(H) is 4-connected if and only if H is essentially 4-edge connected. Conjecture C Every essentially 4-edge connected graph contains a dominating circuit.

37 4-edge Connectivity is a Stronger Condition Note that 4-edge connected graphs contain two edge-disjoint spanning trees.

38 4-edge Connectivity is a Stronger Condition Note that 4-edge connected graphs contain two edge-disjoint spanning trees. Hence, 4-edge connected graphs contain dominating circuits.

39 4-edge Connectivity is a Stronger Condition Note that 4-edge connected graphs contain two edge-disjoint spanning trees. Hence, 4-edge connected graphs contain dominating circuits. Hence, line graphs of 4-edge connected graphs are hamiltonian.

40 Cubic Graphs Definition A graph H is cyclically 4-edge-connected if H contains no edge-cut R such that R < 4 and at least 2 components of H R contain a cycle.

41 Cubic Graphs Definition A graph H is cyclically 4-edge-connected if H contains no edge-cut R such that R < 4 and at least 2 components of H R contain a cycle. Remark A cubic graph is essentially 4-edge-connected if and only if it is cyclically 4-edge-connected.

42 Ash and Jackson Cubic Graph Conjecture Conjecture D Every cyclically 4-edge-connected cubic graph has a dominating cycle.

43 Ash and Jackson Cubic Graph Conjecture Conjecture D Every cyclically 4-edge-connected cubic graph has a dominating cycle. Theorem (Fleischner and B. Jackson 1989) Conjecture D is equivalent to conjectures A,B, and C.

44 Fleischner s Conjecture Conjecture E Every cyclically 4-edge-connected cubic graph that is not 3-edge-colorable has a dominating cycle.

45 Fleischner s Conjecture Conjecture E Every cyclically 4-edge-connected cubic graph that is not 3-edge-colorable has a dominating cycle. Theorem (Kochol 2000) Conjecture E is equivalent to Conjectures A,B,C, and D.

46 Snark Conjecture Definition A snark is a cyclically 4-edge-connected cubic graph of girth at least 5 that is not 3-edge-colorable.

47 Snark Conjecture Definition A snark is a cyclically 4-edge-connected cubic graph of girth at least 5 that is not 3-edge-colorable. Conjecture F Every snark has a dominating cycle.

48 Snark Conjecture Definition A snark is a cyclically 4-edge-connected cubic graph of girth at least 5 that is not 3-edge-colorable. Conjecture F Every snark has a dominating cycle. Theorem (Broersma, Fijavz, Kaiser, Kužel, Ryjáček and Vrana, 2008) Conjectures A F are equivalent.

49 Stronger Appearing Equivalent Conjectures Conjecture G Every 4-connected claw-free graph is hamiltonian connected.

50 Stronger Appearing Equivalent Conjectures Conjecture G Every 4-connected claw-free graph is hamiltonian connected. Conjecture H Every 4-connected line graph is hamiltonian connected.

51 Stronger Appearing Equivalent Conjectures Conjecture G Every 4-connected claw-free graph is hamiltonian connected. Conjecture H Every 4-connected line graph is hamiltonian connected. Theorem (Kužel, Ryjáček, Vána, Xiong)

52 Stronger Appearing Equivalent Conjectures Conjecture G Every 4-connected claw-free graph is hamiltonian connected. Conjecture H Every 4-connected line graph is hamiltonian connected. Theorem (Kužel, Ryjáček, Vána, Xiong) 1 Conjectures G and H are equivalent.

53 Stronger Appearing Equivalent Conjectures Conjecture G Every 4-connected claw-free graph is hamiltonian connected. Conjecture H Every 4-connected line graph is hamiltonian connected. Theorem (Kužel, Ryjáček, Vána, Xiong) 1 Conjectures G and H are equivalent. 2 Conjectures A H are equivalent.

54 1-Hamiltonian-Connected Definition A graph G is 1-Hamilton-connected if for any vertex x of G there is a Hamilton path in G x between any two vertices.

55 1-Hamiltonian-Connected Definition A graph G is 1-Hamilton-connected if for any vertex x of G there is a Hamilton path in G x between any two vertices. Conjecture I Every 4-connected claw-free graph is 1-Hamiltonian Connected.

56 1-Hamiltonian-Connected Definition A graph G is 1-Hamilton-connected if for any vertex x of G there is a Hamilton path in G x between any two vertices. Conjecture I Every 4-connected claw-free graph is 1-Hamiltonian Connected. Theorem (Ryjáček, Saburov, and Vána, 2011) Conjectures A I are equivalent.

57 Survey Paper How Many Conjectures can you Stand? a Survey

58 Survey Paper How Many Conjectures can you Stand? a Survey by Broersma Ryjáček, and P. Vrána