Graph_Theory_Notes2.pdf

# Solved in 1736 by leonhard euler model by a

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Solved in 1736 by Leonhard Euler. Model by a multigraph. Problem becomes: Is there a closed trail containing all the edges? No such trail exists in this multigraph. Why? Eulerian multigraphs Definition 1. An eulerian trail of a multigraph is a trail containing all the edges. An eulerian circuit is a closed eulerian trail (i.e. the first and last vertices are the same). A multigraph that contains an eulerian circuit is called eulerian . Euler’s Theorem Theorem 7 (Euler) . A connected multigraph G is eulerian if and only if the degree of each vertex of G is even. The proof leads to an algorithm for finding eulerian circuits. Algorithm for finding an eulerian circuit Input: multigraph G such that all vertices are even Output: an eulerian circuit of G . 1. Starting with any vertex v , walk on the graph as far as possible without traversing an edge twice. This trail T has to terminate at v and hence is a closed trail. 2. If G - E ( T ) has no edge, then T is an eulerian circuit of G and so we stop. Otherwise, go to Step 3.

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3. Choose any vertex w in T with degree 1 in G - E ( T ). (Since G is connected such a vertex exists.) Starting with w , apply the algorithm in Step 1 to obtain a closed trail A in the component of G - E ( T ) containing w . Replace w by A in T to get a longer closed trail T * . 4. Set T := T * and return to Step 2. Eulerian trails Theorem 8. A connected multigraph G that is not eulerian has an eulerian trail if and only if G has exactly two odd vertices. Moreover, every eulerian trail begins at one odd vertex and ends at the other odd vertex. The proof gives an algorithm for finding an eulerian trail that is similar to the previous algorithm. The only difference is that in Step 1 we should start with an odd vertex v , and T must be an open trail that ends at the other odd vertex. 6 Distance in graphs Distance Definition 5. The distance between a pair of vertices u and v in a graph G is the length of a shortest path joining u and v . If G has no ( u, v )-path, the distance between u and v is defined to be . This distance is denoted by dist G ( u, v ), or dist( u, v ) if it is clear what G is. Theorem 9. The distance function on a graph is a metric ; i.e., it satisfies (a) dist( u, v ) 0, and dist( u, v ) = 0 iff u = v . (b) dist( u, v ) = dist( v, u ) for all u, v V ( G ). (c) dist( u, v ) dist( u, w ) + dist( w, v ) for all u, v, w V ( G ) ( triangle in- equality ). Proof of (c). Let P be a shortest ( u, w )-path and Q a shortest ( w, v )-path. The concatenation of P and Q (i.e. P followed by Q ) is a ( u, v )-walk W . W contains a ( u, v )-path P * . Let ( H ) denote the length of a walk H . Then dist( u, v ) ( P * ) ( W ) = ( P ) + ( Q ) = dist( u, w ) + dist( w, v ) . Diameter and radius Definition 6. The eccentricity ecc( v ) of a vertex v in a graph G is the maximum distance between v and any vertex of G . That is, ecc( v ) = max { dist( v, u ) : u V ( G ) } .
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