Valentin Polishchuk Helsinki Institute for Information Technology, University of Helsinki

Joint work with Esther Arkin and Joseph Mitchell

Applied Math and Statistics, Stony Brook University

Two New Classes of Hamiltonian Graphs

Induced Graph

Subset S of R2

– vertices: S

– edge (i,j) if |i – j | = 1

Square Grid Graph

• Subset S of Z2

• Solid grid– no “holes”– all bounded faces –

unit squares

Hamiltonicity of Square Grids

• NP-complete in general [Itai, Papadimitriou, and Szwarcfiter ’82]

• Solid grids– polynomial

[Umans and Lenhart ’96]

Tilings

• Square grid– unit squares

Tilings

• Square grid– unit squares

• Triangular grid– unit equilateral triangles

Triangular Grid Graph

Subset S

vertices: S– edge (i,j) if

|i – j | = 1

Hole:bounded face ≠ unit equilateral ∆

Solid Triangular Grid

No holesall bounded faces –

unit equilateral triangles

Previous Work

• HamCycle Problem– NP-complete in general

• Solid grids– always Hamiltonian

• no deg-1 vertices

The only non-Hamiltonian solid triangular grid

Local CutSingle vertexwhose removaldecreases number of holes

Solid ) No local cuts

Our result:Triangular grids without

local cuts are Hamiltonian

Idea

• B:– Cycle around the outer boundary– Cycles around holes’ boundaries

• Use modifications– cycles go through all internal vertices

• Exists “facing” rhombus– no local cuts = graph is “thick”– merge facing cycles

• Decrease number of cycles• Get Hamiltonian Cycle

L-modification

V-modification

Z-modification

Priority: L , V , Z

• L

• V

• Z

Wedges

• Sharp– 60o turn

• Wide– 120o turn

The Main Lemma

Until B passes through ALL internal vertices– either L, V, or Z may be applied

small print:unless G is the Star of David

Internal vertex v not in B

• A neighbor u is in B

• Crossed edges – not in B– o.w. – apply L

How is u visited?

WLOG, 1 is in B

s is in BL cannot be applied

s

How is s visited?

Sharp Wedge

Z

Vs

s

Wide Wedge

L cannot be applied t is in B

Deja Vu

s

Rhombus– edge of B– vertex not in B– vertex in B

Unless – t is a wide wedge

• modification!• welcome new vertex to B

Another Wide Wedge

Yet Another vertex– Yet Another rhombus

Yet Another wide wedge

And so on…

Star of David!

Cycle Cover → HamCycle

• Cycles around the outer boundary• Cycles around holes’ boundaries• Use modifications

– cycles go through all internal vertices

• Exists “facing” rhombus– no local cuts = graph is “thick”– merge facing cycles

• Decrease number of cycles• Get Hamiltonian Cycle

Hamiltonian Cycles in

High-Girth Graphs

HamCycle Problem is NP-complete

• Classic • Girth?

– 4 [GJ]

– 3 [CLRS]

• NP-complete [Garey, Johnson, Tarjan’76]

– planar– cubic– girth-5

Higher girth?

Multi-Hamiltonicity• 1 HC 2 HCs

cubic [Smith], any vert – odd-deg [Thomason’78]

r-regular, r > 300 [Thomassen’98], r > 48 [Ghandehari and Hatami]

4-regular? conjecture [Sheehan’75]

maxdeg ≥ f( maxdeg/mindeg ) [Horak and Stacho’00]

bipartite, mindeg in a part = 3 [Thomassen’96]

• 1 HC exp(maxdeg) HCs [Thomassen’96]

– bipartite

• 1 HC exp(girth) HCs [Thomassen’96]

cubic or bipartite, mindeg in a part = 4

Planar maxdeg 3, high-girth?

>1 HC? Small # of HCs?

Our Contribution

Planar

maxdeg 3

arbitrarily large girth

• HamCycle Problem is NP-complete

• Exactly 3 HamCycles arbitrarly large # of vertices

The Other Tiling: Infinite Hexagonal Grid

• Induced graphs– hexagonal grids

Is HamCycle Problem NP-hard for hexagonal grids?

Attempt to Show NP-Hardness• Same idea as for square and triangular grids

[Itai, Papadimitriou, and Szwarcfiter ‘82, Papadimitriou and Vazirani ’84, PAM’06]

• HamCycle Problem– undirected planar bipartite graphs

– max deg 3

G0

Embed

0o, 60o, 120o segments

(Try to) Embed in Hex Grid

Edges – Tentacles

Traversing Tentacles

Cross pathconnects adjacent nodes

Return path returns to one of the nodes

White Node Gadget

Middle Vertex: 2 edges…

Middle Vertex: 2 edges…

Induces 2 cross, 1 return path

Induces 2 cross, 1 return path

Induces 2 cross, 1 return path

Black Node Gadget

Middle Vertex: 2 edges…

Middle Vertex: 2 edges…

Induces 2 cross, 1 return path

Induces 2 cross, 1 return path

Induces 2 cross, 1 return path

Return Path Starts at white node Closes at black node

HC in G HC in G0

Any node gadgetadjacent to

2 cross paths

1 return path

• Edges of G0 in HC

Cross paths

• Edges of G0 not in HC

Return paths from white nodes

No… didn’t show how to turn a tentacle

Can’t turn with these tentacles

Ham Cycle is NP-hard for Hex Grid?

No Longer in a Hex Grid

Subdivide (Shown) Edges

Imagine:adjacent deg-2 vertices

connected by length-g path

Girth g

Girth g+2 Graph

• Planar– turning tentacle

• no longer an issue– not in a hex grid

• Maxdeg 3

HC in G HC in G0

Any node gadgetadjacent to

2 cross paths

1 return path

• Edges of G0 in HC

Cross paths

• Edges of G0 not in HC

Return paths from white nodes

Theorem 1

For any g ≥ 6

HamCycle is NP-hard inplanar

deg ≤ 3non-bipartite

girth-ggraphs

Multi-Hamiltonicity

• Planar

• Bipartite

• Maxdeg 3

Exactly 3 HamCycles

Theorem 2

For any g ≥ 6

existsplanar

deg ≤ 3non-bipartite

girth-ggraph

with exactly 3 HamCycles

Summary

• Trangular grids no local cut ) Hamiltonian

• maxdeg-3 planar girth-g– HamCycle Problem is NP-complete– exists graphs with exactly 3 HamCycles

Open

• HamCycle Problem in hexagonal grids