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Published byOswaldo Wamsley Modified over 2 years ago

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Graph Colouring The Team : Aymen Dammak Sébastien Jagueneau Florian Lajus Xavier Loubatier Cyril Rayot Mathieu Rey Mentor : Paul-Yves Gloess 1

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First Definitions A graph is a set of vertices linked by edges A graph 2

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Neighbours A vertex is a neighbour of another vertex if they are linked by an edge 3

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Colouring Notions A colour is associated to each vertex A graph is well-coloured if no neighbouring vertices share the same colour A well-coloured graph 4

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Degree Notions The vertex degree is the number of neighbours of this vertex The graph degree is the maximum degree of its vertices 5

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The Theorem Choose randomly a vertex H which has all its neighbours well-coloured Choose randomly a color C unused by the neighbours of H Color H with C and graph is still well- coloured

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What is PVS ? PVS : Prototype Verification System Proof assistant SRI international

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Divide and Conquer Type Checking -> TCC : Type Correctness Condition Corrections Definition of degree Graph non oriented After correction, 3 TCCs required to be proved Coloring_vertex_TCC1 Coloring_vertex_TCC2 Coloring_TCC5

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Proof of coloring_vertex_tcc1 Main idea : nonempty (difference (below (1+degree (R)), image (f, neighbours (R) (t))) below (1+degree (R)): N+1 colours of the graph image (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. 9

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Proof of coloring_vertex_tcc1 Graph colours: 10

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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. 11

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Proof of coloring_vertex_tcc1 Proof idea: If a set has more elements than another one, the difference between these two sets is not empty. The notion of cardinality Three new lemmas to prove the tcc1 12

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Proof of coloring_vertex_tcc1 Encountered problems: Not acquainted with the PVS syntax Too strong hypothesis in one of the three lemmas not easy to prove 13

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Proof of coloring_vertex_tcc2 Main idea : a well coloured graph can be created by adding a coloured vertex to a well coloured graph. 14

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Proof of coloring_vertex_tcc2 A well-coloured graph 15

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Proof of coloring_vertex_tcc2 A vertex is selected 16

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Proof of coloring_vertex_tcc2 An unused colour is selected 17

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Proof of coloring_vertex_tcc2 The coloured vertex is added to the well-coloured graph 18

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Proof of coloring_vertex_tcc2 Two steps : Modification of the colouring function. Modification of the well-coloured graph. 19

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Proof of coloring_tcc5 Main 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 20

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Proof of coloring_tcc5 A subset, a strict subset and an element Yellow area : vertices remaining uncoloured X 21

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Proof of coloring_tcc5 Adding the element to the subset Yellow area becomes the green one X 22

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Proof of coloring_tcc5 Green area strict subset of the yellow area X X 23

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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. 24

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Conclusion Proofs are complete Using PVS Think whether what we are writing is correct and why Step by step correction Mathematical logic Finishing the project 25

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Thanks for your attention 26

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