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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