Line graphs of and

Hamilton-connectedness of claw-free graphs

ZdenˇekRyj´aˇcek,Plzeˇn

Joint work with Petr Vr´ana,Plzeˇn

1 Graph: no loops, no parallel edges : no loops, parallel edges possible c(G): the circumference of G (the length of a longest in G) cl(G): the closure of G

Theorem [1997]. Let G be a claw-free graph. Then (i) cl(G) is uniquely determined, (ii) c(cl(G)) = c(G), (iii) cl(G) is the of a triangle-free graph.

Corollary. G is hamiltonian if and only if cl(G) is hamiltonian.

The closure operation cl(G): • turns a claw-free graph into a line graph • preserves hamiltonicity or non-hamiltonicity

...... •...... •.. ..•...... •.. •...... •• ...... ••...... •...... • ...... •...... •. •• ...... •...... • ...... •......

2 An immediate application:

Theorem [Zhan, 1991]. Every 7-connected line graph (of a multigraph) is Hamilton-connected.

From this:

Theorem [1997]. Every 7-connected claw-free graph is hamiltonian.

Theorem [Hu, Tian, Wei 2005]. Every 6-connected line graph (of a multigraph) with at most 29 vertices of 6 is Hamilton-connected.

Corollary. Every 6-connected claw-free graph with at most 29 vertices of degree 6 is hamiltonian.

Hamilton-connectedness:

Theorem [Brandt, 1999]. Every 9-connected claw-free graph is Hamilton- connected.

Theorem [Hu, Tian, Wei, 2005]. Every 8-connected claw-free graph is Hamilton-connected.

3 C – a class of graphs

We say that C is stable if G ∈ C ⇒ cl(G) ∈ C.

Examples: k-connected claw-free graphs chordal claw-free graphs

P – a property C – a stable class

We say that P is stable in C if, for any G ∈ C, G has P ⇔ cl(G) has P.

Example: hamiltonicity is a stable property in the class of k-connected claw-free graphs.

π – a graph invariant C – a stable class

We say that π is stable in C if, for any G ∈ C, π(G) = π(cl(G)).

Example: circumference is a stable invariant in the class of k-connected claw-free graphs.

Which properties are stable?

4 Property / invariant Stable Connectivity Circumference YES 1 Hamiltonicity YES 1 Length of a longest YES 1 Traceability YES 1 NO 3 Homogeneous traceability ??? 4 ≤ κ ≤ 6 YES 7 () pancyclicity NO any κ ≥ 2 (Full) cycle extendability NO any κ ≥ 2 Having a 2-factor with ≤ k components YES 1 Minimum number of components in a 2-factor YES 1 Having a cycle cover with ≤ k cycles YES 1 Minimum number of cycles in a cycle cover YES 1 Having a path factor with ≤ k components YES 1 [Ishizuka] Minimum number of components in a path factor YES 1 Having a path cover with ≤ k paths YES 1 [Ishizuka] Minimum number of paths in a path cover YES 1 Hamiltonian index YES 1 Having hamiltonian prism YES 1 [Cada]ˇ NO 1 Having a P3-factor YES 2 [Kaneko et al.] NO 3 Hamilton-connectedness ??? 4 ≤ κ ≤ 7 YES 8 [Hu,Tian,Wei] NO 3 Hamilton-connectedness ??? 4 ≤ κ ≤ 6 YES 7 [ZR, Vr´ana,2008]

5 pa(G) - the length of a longest path with one endvertex at a (given) vertex a ∈ V (G)

Homogeneous traceability (having a with one endvertex at a for every a ∈ V (G))

...... •...... •••...... •..... x . •••...... •••...... •......

NOT STABLE

  Homogeneous traceability  is not stable in 2-conected claw-free graphs.  pa(G) 

6 . . . . •. •...... H ...... •...... •...... •...... •...... •...... •. .. .. • ...... •. .. .. • ...... •• ...... • • ...... ••...... •...... •. •...... •...... •...... • •...... • ...... •...... •...... •...... •...... • ...... • ...... ••• ...... •. •• ...... e...... •• ...... • ...... •...... •...... • ...... •...... • ...... •• ...... •• ...... • ...... • ...... •...... •. . . •. G = L(H) is 3-connected, not homogeneously traceable cl(G) is homogeneously traceable

  Homogeneous traceability  is not stable in 3-conected claw-free graphs.  pa(G) 

BUT: every 7-connected claw-free graph is hamiltonian ⇒ every 7-connected claw-free graph is homogeneously traceable ⇒   Homogeneous traceability  IS STABLE 7-conected claw-free graphs.  pa(G) 

Unstable properties can be made stable by restricting the class of graphs under consideration.

7 pab(G) - the length of a longest (a, b)-path (for given a, b ∈ V (G)) Hamilton-connectedness

. . •...... •...... G = L(H) ...... H : ...... •...... •...... • ...... •• ...... ••...... • ...... •...... •...... • ...... •• ...... • ...... •...... • ...... •• ...... x...... •• ...... y ......

G = L(H) is Hamilton-connected ⇐⇒ H contains an internally dominating (x, y)-trail for any x, y ∈ E(G)

G = L(H) is 3-connected, not Hamilton-connected cl(G) is Hamilton-connected

  Hamilton-connectedness  is not stable in 3-conected claw-free graphs.  pab(G) 

BUT: every 8-connected claw-free graph is Hamilton-connected ⇒

  Hamilton-connectedness  IS STABLE in 8-conected claw-free graphs.  pab(G) 

8 k-closure x ∈ V (G) is locally k-connected if N(x) induces a k-connected graph. clk(G): local completions only at locally k-connected vertices.

Theorem [Bollob´as,Riordan, ZR., Saito, Schelp, 1999].

(i) clk(G) is uniquely determined for every k ≥ 1.

(ii) Homogeneous traceability is stable under cl2(G).

(iii) Hamilton-connectedness is stable under cl3(G).

Conjecture [Bollob´as,Riordan, ZR., Saito, Schelp, 1999].

Hamilton-connectedness is stable under cl2(G).

A. Kelmans: On graph closures. Discrete Math. 271 (2003), 141-168

Claims to give infinitely many counterexamples to the conjecture

WRONG

9 Example.

...... y a ...... •...... •...... G: ...... • ...... • ...... x ...... • ...... • ...... • ...... •...... •...... b ......

• hN(x)i 2-connected

• No hamiltonian (a, b)-path

0 • There is a hamiltonian (a, b)-path in Gx.

  “Having a hamiltonian (a, b)-path”  is not stable under cl (G).  2 pab(G) 

0 BUT: Gx has no hamiltonian (a, y)-path. 0 Thus: Gx is NOT Hamilton-connected. The example does NOT disprove the conjecture.

Theorem [ZR., Vr´ana,2008].

Hamilton-connectedness is stable under cl2(G).

10 G is 2-closed if G = cl2(G) What is the structure of 2-closed graphs?

...... •...... Not a line graph: ...... •• ...... •• ...... •......

Theorem [Beineke, 1969]. A graph G is a line graph (of some graph) if and only if G does not contain a copy of any of the following graphs as an .

...... • .....•...... •...... •...... ••...... •..... •...... • ...... • ...... • ...... •...... •...... •......

...... •. •...... •...... •...... •...... •...... ••...... • ••...... ••...... • ...... • ...... •...... •...... •......

...... •. •...... •. ..•...... •...... •...... •...... • •...... •...... •...... • ...... • ...... •...... ••...... •...... •......

11 Line graph of a multigraph?

...... Line graphs ...... Line graphs ...... Claw-free graphs ...... of multigraphs......

Theorem [Hemminger 1971; Bermond, Meyer, 1973]. A graph G is a line graph of a multigraph if and only if G does not contain a copy of any of the following graphs as an induced subgraph.

...... ••...... ••...... •...... •...... G ...... G ...... G ...... 1 ...... 2 ...... 3...... G4 ...... •. ••...... ••...... • ...... ••••...... • •• •• ...... •......

...... •...... •...... •...... G5 ...... G6 ...... G7 ...... •...... ••...... ••••...... •...... •...... ••• ...... • ...... • .... • ...... •• ......

IND Lemma. Let G be a claw-free graph, x ∈ V (G), let H ⊂ hN(x)iG be a 2-connected graph containing two distinct pairs of independent vertices.

Then hN(x)iG is 2-connected.

Lemma. Every 2-closed claw-free graph is {G1,G3,G5,G6,G7}-free.

12 Thus, in cl2(G), only G2 or G4 can remain.

...... •...... •...... •...... •...... G ...... G ...... 2 ...... 4 ...... ••...... ••...... • • .... • • ......

“Multigraph closure” clM (G) of a graph G:

Recursive performing cl2(G) and closing specified vertices in copies of G2 or

G4, as long as there is something to do.

J = u0u1 . . . uk+1 – a walk in G. We say that J is good in G, if • k ≥ 4, • J 2 ⊂ G,

• for any i, 0 ≤ i ≤ k − 4, h{ui, ui+1, . . . , ui+5}iG ' S1 or S2.

u1 u3 uk−3 uk−1 ...... •...... •...... •...... •...... u ...... u 0 • ...... • k+1 ...... •...... •...... •...... •...... u2 u4 uk−2 uk

u3 Similarly – a good cycle ...... •...... u ...... u..... = u 5 ...... 1 k+1 ...... •...... •...... • •...... u u ...... 4 2 ...... u.... u ...... k−3 k−1...... •...... •...... •...... •...... u ...... u = u k−4 ...... k 0 ...... •...... uk−2

13 Lemma. Let G be a 2-closed claw-free graph and J = u0u1 . . . uk+1 a good walk in G, k ≥ 5. Then dG(ui) = 4, i = 3, . . . , k − 2.

Lemma. Let G be a connected 2-closed claw-free graph and let C ⊂ G be a good cycle in G, |V (C)| ≥ 6. Then G = C2.

In the rest we suppose that G is not the square of a cycle.

Let J be a good walk in G. We say that J is maximal if, for every good walk J 0 in G, J being a subsequence of J 0 implies J = J 0.

Lemma. Let G be a connected 2-closed claw-free graph that is not the square of a cycle, and let J = u0, u1, . . . , uk+1 be a maximal good walk in

G. Then hNG[u1] \{u3}iG = hNG[u2] \{u3, u4}iG and this subgraph is a .

Lemma. Let G be a connected 2-closed claw-free graph that is not the square of a cycle, and let J = u0u1 . . . uk+1 be a good walk in G. Then u1 . . . uk is a path.

u1 u3 uk−3 uk−1 ...... •...... •...... •...... •...... u u • ...... • k+1 0 ...... •...... •...... •...... •...... u2 u4 uk−2 uk

14 k u1 u3 uk−3 uk−1 J ...... 1 ...... •...... •...... •...... •...... u ...... u 0• ...... • k+1 ...... •...... •...... •...... •...... u2 u4 uk−2 uk

k u1 u3 uk−3 uk−1 J ...... 2 ...... •...... •...... •...... •...... u u •...... •. k+1 0 ...... • • • • ...... u u u u ...... 2 4 k−2 ..k......

k u3 J ...... 3 ...... •...... u ...... u 5 ...... 1 ...... •...... •...... • •...... u u .... . 4 2 ...... u = u . . ...•. 0 k+1 ...... u.... u ...... k−3 k−1...... •...... •...... •...... •...... u ...... u k−4 ...... k ...... •...... uk−2

k u3 J ...... 4 ...... •...... u... = u u ...... 1 k+1 5 ...... •...... •...... • •...... u u ...... 4 2...... u..... uk−1...... k−3 ...... •...... •...... •...... •...... u ...... u = u k−4 ...... k 0 ...... •...... uk−2

15 Let G be a claw-free graph, and let clM (G) be a graph constructed by the following algorithm.

1. Set G1 = cl2(G), i := 1.

2. If Gi contains a good walk, then

(a) choose a maximal good walk J = u0u1 ...k+1, 0 00 (b) let Gi be the local completion of Gi at u1 and Gi be the local com- 0 pletion of Gi at uk, 00 (c) set Gi+1 = cl2(Gi ), i := i + 1, (d) go to (2). M 3. Set cl (G) = Gi. The graph clM (G) will be called (the) M-closure of G.

Theorem. Let G be a claw-free graph. Then (i) clM (G) is uniquely determined, (ii) there is a multigraph H such that clM (G) = L(H), (iii) G is Hamilton-connected if and only if clM (G) is Hamilton-connected.

Hamilton-connectedness is stable under clM (G).

BUT:   “Having a hamiltonian (a, b)-path”  is NOT STABLE under clM (G) !!!  pab(G) 

16 Proposition. A claw-free graph G is M-closed if and only if there is a multigraph H such that G = L(H) and H does not contain a subgraph S (not necessarily induced) with any of the following properties:

(i) S ' T1 and there are u1, u2 ∈ V (H) \ V (S) such that u1 6= u2 and

uixi ∈ E(H), i = 1, 2,

(ii) S ' T2 or S ' T3 and there is a u ∈ V (H) \ V (S) such that |NH(u) ∩

{x1, x2}| = 1,

(iii) S ' T3

(where x1, x2 are the only vertices in S with dS(xi) = 3).

...... x ...... x ...... x ..•. .. ..•...... •...... 1 ...... 1 ...... 1 T ...... T ...... T ...... 1 ...... 2 ...... 3 ...... •...... •• ...... •...... x2 •...... x2 •...... x2

17 Recall: Every 6-connected line graph (of a multigraph) with at most 29 vertices of degree 6 is Hamilton-connected [Hu, Tian, Wei 2005].

Theorem. Every 6-connected claw-free graph with at most 29 vertices of degree 6 is Hamilton-connected.

Corollary. Every 7-connected claw-free graph is Hamilton-connected.

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

Conjecture B. Every 4-connected line graph of a multigraph is Hamilton-connected.

Conjecture C. Every 4-connected claw-free graph is Hamilton-connected.

Theorem [Kuˇzel,Vr´ana,Xiong]. Conjectures A and B are equivalent.

Theorem. Conjectures B and C are equivalent.

18 C – a class of graphs

A closure on C is a mapping cl : C → C such that

• V (G) = V (cl(G))

• E(G) ⊂ E(cl(G))

G = L(H):

. . •...... •...... H ...... •...... •...... • ...... •• ...... ••...... • ...... •...... •...... • ...... •• ...... • ...... •...... • ...... •• ...... x...... •• ...... y ......

Lk – the class of k-connected line graphs (of graphs) M Lk – the class of k-connected line graphs of multigraphs

Theorem [Vr´ana,2008]. M M There is no closure cl on L3 such that cl : L3 → L3 and Hamilton- connectedness is stable under cl.

19