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Graph ColouringThe Team :
Aymen DammakSébastien Jagueneau
Florian LajusXavier Loubatier
Cyril RayotMathieu Rey
Mentor :Paul-Yves Gloess
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First DefinitionsA graph is a set of vertices linked by edges
A graph
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NeighboursA vertex is a neighbour of another vertex if
they are linked by an edge
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Colouring NotionsA colour is associated to each vertexA graph is well-coloured if no neighbouring
vertices share the same colour
A well-coloured graph
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Degree NotionsThe vertex degree is the number of
neighbours of this vertexThe graph degree is the maximum degree of
its vertices
The Theorem• Choose randomly a vertex H which has all its neighbours well-coloured• Choose randomly a color C unused by theneighbours of H• Color H with C and graph is still well-coloured
What is PVS ?PVS : Prototype Verification SystemProof assistant SRI international
Divide and ConquerType Checking -> TCC : Type Correctness
ConditionCorrections
Definition of degreeGraph non oriented
After correction, 3 TCCs required to be provedColoring_vertex_TCC1Coloring_vertex_TCC2Coloring_TCC5
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Proof of coloring_vertex_tcc1Main idea :
nonempty (difference (below (1+degree (R)), image (f, neighbours (R) (t)))
below (1+degree (R)): N+1 colours of the graphimage (f, neighbours (R) (t)): colours of t neighbours
The set of colours used to colour the neighbours of vertex T, can not
include all the graph colours.
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Proof of coloring_vertex_tcc1
Graph colours:
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Proof of coloring_vertex_tcc1
Colours of vertex V neighbours:
There is always a colour left for vertex V, different from the colours of its neighbours.
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Proof of coloring_vertex_tcc1Proof idea:
If a set has more elements than another one, the difference between these two sets is not empty.
The notion of cardinalityThree new lemmas to prove the tcc1
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Proof of coloring_vertex_tcc1
Encountered problems:Not acquainted with the PVS syntaxToo strong hypothesis in one of the three
lemmasnot easy to prove
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Proof of coloring_vertex_tcc2Main idea :
a well coloured graph can be created by adding a coloured vertex to a well coloured graph.
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Proof of coloring_vertex_tcc2A well-coloured graph
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Proof of coloring_vertex_tcc2A vertex is selected
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Proof of coloring_vertex_tcc2An unused colour is selected
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Proof of coloring_vertex_tcc2The coloured vertex is added to the well-
coloured graph
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Proof of coloring_vertex_tcc2Two steps :
Modification of the colouring function.
Modification of the well-coloured graph.
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Proof of coloring_tcc5Main idea :
Set “S” representing the vertices (coloured or not) of a graph
Strict subset “s” of the set (coloured vertices)An element “x” not part of the subset (vertex
about to be coloured)The number of vertices to be coloured is
always decreasing
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Proof of coloring_tcc5A subset, a strict subset and an elementYellow area : vertices remaining uncoloured
X
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Proof of coloring_tcc5Adding the element to the subsetYellow area becomes the green one
X
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Proof of coloring_tcc5Green area strict subset of the yellow area
XX
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Proof Result
In order to compare: last year, at the end of their project, they had 36 proved and 6 unproved lemmas with 11 TCCs for a total of nearly 21 seconds.
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ConclusionProofs are completeUsing PVS
Think whether what we are writing is correct and why
Step by step correctionMathematical logic
Finishing the project
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Thanks for your attention