two new classes of hamiltonian graphs
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Two New Classes of Hamiltonian Graphs. 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. Induced Graph. Subset S of R 2 vertices: S - PowerPoint PPT PresentationTRANSCRIPT
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