Prestation_ClydeShen

GRAPH COLOURING ON
GRAPH DYNAMICIAL SYSTEM
Clyde Shen
Supervisor by: Dr. Jiamou Liu

Overview
• Graph Theory
– Graph
– Graph Colouring
• Automaton
– deterministic finite automaton (DFA)
• Graph Dynamical System
– Graph-cellular Automaton
– Some specific case: linear graph, circle graph, tree
graph, wheel graph and Peterson graph

Graph Theory
Leonhard Paul Euler (1707- 1783) : a
pioneering Swiss mathematician
• Mathematical notation
• Analysis
• Number theory
• Graph Theory
• Applied mathematics
• Physics and Astronomy
• Logic
The Seven Bridges of Königsberg :
Is it possible to take a walk through town,
starting and ending at the same place,
and cross each bridge exactly once?

Graph Theory
Graph 𝐺 = (𝑉, 𝐸)
𝑉 is a finite set of vertices
𝐸 is a finite set of edges
V = {1,2,3,4,5}
E = { {1, 4}, {2, 4}, {3, 5},
{4, 5}, {2, 5} }

Graph Theory
Graph 𝐺 = (𝑉, 𝐸)
𝑉 is a finite set of vertices
𝐸 is a finite set of edges
Undirected Graph Directed Graph

Graph Theory
• Undirected Graph
• Planar Graph
– a graph can be drawn on a plane surface
– No two edges intersect

Graph Theory
• Undirected Graph
• Planar Graph
– a graph can be drawn on a plane surface
– No two edges intersect
• Graph Colouring

Graph Theory
Graph Colouring Problem:
• 𝐺 = (𝑉, 𝐸)
• set 𝐶 is a function 𝑓 ∶ 𝑉 → 𝐶, where 𝐶 is a
set of colors
• ∀𝑖,𝑗 : 𝑣𝑖, 𝑣𝑗 ∈ 𝐸 ⇒ 𝑓(𝑣𝑖) ≠ 𝑓(𝑣𝑗)

Graph Theory
𝐶 = 4
𝐶 = 2

Graph Theory
• The chromatic number of a graph 𝐺 is the size
of a smallest set 𝐶 for there is a proper
colouring of G with 𝐶
• denoted by 𝜒(𝐺)
• If 𝜒 𝐺 ≤ 𝑘 , 𝐺 is k-colourable

Graph Theory
• Computational complexity
– Graph colouring is NP-complete
– NP-hard to compute the chromatic number
– Goal: design efficient algorithm for finding
colouring using ≥ 𝜒(𝐺) colours

Graph Theory
• Algorithms for colouring

Graph Theory
• Algorithms for colouring
– Greedy colouring
http://en.wikipedia.org/wiki/Greedy_coloring
– Parallel and distributed algorithms
Schnitger, G. (2009). Parallel and Distributed Algorithms. Algorithms, 10.

Automaton
• Deterministic finite automaton (DFA)
– DFA is a 5-tuple (𝑄, 𝛴, 𝛿, 𝑞0 , 𝐹)
• Q is a finite set of states
• 𝛴 is a finite set of input symbols called the alphabet
• 𝛿 is transition function (𝛿: 𝑄 × 𝛴 → 𝑄)
• 𝑞0 is start state(𝑞0 ∈ 𝑄)
• 𝐹 is set of accepting states (𝐹 ⊆ 𝑄)

Automaton
• Deterministic finite automaton (DFA)
– DFA is a 5-tuple (𝑄, 𝛴, 𝛿, 𝑞0 , 𝐹)
– Example of a DFA 𝑀 = (𝑄, 𝛴, 𝛿, 𝑞0 , 𝐹)
• 𝑄 = {𝑆1, 𝑆2}
• 𝛴 = {0,1}
• 𝑞0 = 𝑆1
• 𝐹 = {𝑆1}
• 𝛿 is defined by the following state transition table:
0 1
𝑆1 𝑆2 𝑆1
𝑆2 𝑆1 𝑆2

Graph Dynamical System
• Let a graph 𝐺 = 𝑉, 𝐸 be a undirected graph
– ∀ 𝑣 ∈ 𝑉, denote the set of neighbours by ℕ(v)
– Vertex has no more than 𝑑 𝑔 neighbours.
• A local orientation ∆ is a function
𝜇 ∶ 𝑉 × {1,2, … , 𝑑 𝑔} → (𝑉 ∪ {□}) where □ ∉𝑉
∀ 𝑣 ∈ 𝑉 we have rules:

• Graph-cellular Automaton (GA)

𝔄 is a 4-tuple 𝔄 = (𝑄, 𝑞0, ∆, 𝜇) consisting of
– Q is a finite set of states.
– q0 is a quiescent state.
– ∆: 𝑄 × (𝑄 ∪ {□}) 𝑑 𝑔→ 𝑄 is the local transition.
– 𝜇 is a local orientation of graph

𝔄 is a 4-tuple 𝔄 = (𝑄, 𝑞0, ∆, 𝜇) consisting of
– Q is a finite set of states.
– q0 is a quiescent state.
– ∆: 𝑄 × (𝑄 ∪ {□}) 𝑑 𝑔→ 𝑄 is the local transition.
– 𝜇 is a local orientation of graph
quiescent state : ∆ 𝑞0, 𝑞1, 𝑞2, … , 𝑞 𝑑 𝑔
= 𝑞0 if
𝑞1, 𝑞2, … , 𝑞 𝑑 𝑔
∈ {𝑞0, □}

Dead vertex Alive vertex Alive vertex

• Basic local transition rule:
– ∀ 𝑣∈ 𝑉: ∆ 𝑞 𝑣, 𝑞1, 𝑞2, … , 𝑞 𝑑 𝑔
= 𝑞′
𝑣 where
𝑞𝑞 𝑣 = min(𝑞 𝑣, 𝑞𝑞 𝑣)

• Basic local transition rule:
– ∀ 𝑣∈ 𝑉: ∆ 𝑞 𝑣, 𝑞1, 𝑞2, … , 𝑞 𝑑 𝑔
= 𝑞′
𝑣 where
𝑞𝑞 𝑣 = min(𝑞 𝑣, 𝑞𝑞 𝑣)
(1)
(2)

For example:
• Linear graph

For example:
• Linear graph
– Odd number of vertex
– Even number of vertex

Odd number of vertex:

Done!

Even number of vertex:

Even number of vertex:
Done!

• Linear graph 

• Circle graph
– Even number of vertex
– Odd number of vertex

Loop!

• Circle graph 
• Tree graph

Tree graph:

Tree graph:
Done!

• Tree graph 
• Wheel graph

Wheel Graph:

Wheel Graph:
Loop!

• Tree graph 
• Wheel graph 
• Peterson graph

Peterson graph:

Peterson graph:
Loop!

• Tree graph 
• Wheel graph 
• Peterson graph 

Conclusion
Result:
INPUT:
OUTPUT:
Graph-cellular Automaton
+ Rules

Conclusion
• Graph Theory
– Graph: planar graph
– Graph Colouring: chromatic number
• Automaton
– deterministic finite automaton (DFA)
• Graph Dynamical System
– Graph-cellular Automaton
– Some specific case: linear graph, circle graph, tree
graph, wheel graph and Peterson graph

Conclusion
• Initialize the use of graph dynamical system
(graph-cellular automaton)
• Graph-cellular automaton can be used as a
viable tool in obtaining a colouring for specific
classes of graph
• Extend this work to more general classes of
graph

Prestation_ClydeShen

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