a Find a subset of the vertices that, together with the edges that directly connect them, form a subgraph that is K . 5
b What can you conclude from this about the original graph? Two of these graphs are isomorphic. 1 A B C D E
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a Which two graphs are isomorphic?
b Show that the two graphs from your answer to part a are isomorphic.
A simple-connected Eulerian graph is drawn that has exactly nine vertices and 12 edges.
SA
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c Show that each of the other two graphs are is isomorphic to any of the other graphs.
a Write down the minimum possible vertex degree. b What is the maximum number of vertices that can have this minimum degree? c What is the maximum number of vertices of degree 4 that the graph could have?
9
A three-dimensional noughts and crosses game is played using a 3 × 3 × 3 cube of cells. Label the cells (a, b, c), where a, b, c ∈ {1, 2, 3} represent the coordinates of the cell. A simple-connected graph is constructed in which the vertices represent the cells and edges join two cells if those cells are not in the same straight line of 3 cells.
a Which cells are directly connected to (1, 2, 3)? b Show that the graph is Eulerian. 10
A graph has exactly four vertices. The degrees of the vertices are 2, 2, 4 and x. Original material © Cambridge University Press 2018