Importance of odd and even numbers.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
We can trace over the edges exactly once in diagrams (i), (ii), (iii) and (vii).
We cannot trace over the edges exactly once in diagrams (iv), (v) and (vi).
Let us analyze why it is not traceable (or traversable).
In figure (i) A, B, C and D are called vertices.
Like these the vertices in diagrams(ii) to (vii) are as follows:
(ii) P, Q, R, S (iii) K, L, M, N, O (iv) E, F, G, H, I (v) J, K,L, M, N (vi) S, T, U, V, W (vii) I, J, K, L, M.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In figure (i), AB and AD meet at A. Hence, A is an even vertex,
In figure (i), there are 4 even vertices (all are even vertices).
In figure (i), we can start at any vertex and we end at the same vertex. It is traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In figure (ii), QP, RP, SP meet at P. Hence, P is an odd vertex.
In figure (ii), P and R odd vertices. Q and S are even vertices.
In figure (ii) we have to start at anyone of the odd vertices P or R.
We end in the opposite vertex (ii) (starting point P and ending in R or starting with R and ending in P). It is traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In fig (iii), K and L are odd vertices. M, O, N are even vertices.
In figure (iii) we have to start at anyone of the odd vertices K or L.
In (iii), Point of start K end point L. Point of start L, end K.
It is traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In fig. (iv), F is the only even vertex. E, I, G, H are odd vertices.
In figure (iv) there are 4 odd vertices, We find the diagram is not traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In fig. (v), K is the only even vertex. J, L, M, N are odd vertices.
In figure (v) there are 4 odd vertices, We find the diagram is not traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In figure (vi) at W, SW, TW, UW, VW meet at W. Hence, it is an even vertex
In fig. (vi),S, T, U, V are odd vertices and W is the only even vertex.
In figure (vi) there are 4 odd vertices, We find the diagram is not traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In (vii), I, J, K, L and M are all even vertices.
Further, it is traversable.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In figure (i), AB and AD meet at A. Hence, A is an even vertex,
In figure (ii), QP,RP, SP meet at P. Hence, P is an odd vertex.
In figure (vi) at W, SW, TW, UW, VW meet at W. Hence, it is an even vertex
In figure (i), there are 4 even vertices (all are even vertices).
In figure (ii), P and R odd vertices. Q and S are even vertices.
In fig (iii), K and L are odd vertices. M, O, N are even vertices.
In fig. (iv), F is the only even vertex. E, I, G, H are odd vertices.
In fig. (v), K is the only even vertex. J, L, M, N are odd vertices.
In fig. (vi),S, T, U, V are odd vertices and W is the only even vertex.
In (vii), I, J, K, L and M are all even vertices.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
In figure (i), we can start at any vertex and we end at the same vertex. It is traversable.
In figures (ii) and (iii) we have to start at anyone of the odd vertices P or R.
We end in the opposite vertex (ii) (starting point P and ending in R or starting with R and ending in P).
In (iii), Point of start K end point L. Point of start L, end K.
In figures (iv), (v) and (vi) there are 4 odd vertices, We find these diagrams are not traversable.
In figure (vii), all are even vertices. Further, it is traversable.
These diagrams are called NETWORKS.
Without removing the pencil from the paper and without tracing an edge more than once (traversable) can we draw the diagrams?
Now, we can draw the following conclusions:
(i) A network with no odd (or all even) vertices is traversable. We may start from any vertex and we will end where we began.
(ii) A network with exactly 2 odd vertices is traversable. We must start at either of the odd vertices and finish at the other.
(iii) A network with more than 2 odd vertices, is not traversable.
From this, we can see the importance of odd and even numbers.
Find out whether Traversable or not? If it is traversable draw without removing the pencil & without tracing edge more than once
Find out whether Traversable or not? If it is traversable draw without removing the pencil & without tracing edge more than once
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